How do Buddhists reconcile "Anatta" with Buddha supporting the existence of the Self in the Mahayana Mahaparininirvana Sutra? In the third chapter of the Mahayana Mahaparininirvana Sutra, Buddha calls the Self real and permanent:


  Then the Buddha said to all the bhiksus: "Do not say this. I now leave all the unsurpassed Dharma in the hands of Mahakasyapa. This Kasyapa will henceforth be the one upon whom you may rely. This is as in the case where the Tathagata becomes the one to whom all beings can turn. The same is the case with Mahakasyapa. He will now become your refuge. This is as in the case of a king who has many territories and who goes on a tour of inspection, leaving all affairs of state in the hands of his minister. The same with the Tathagata. All right teachings are left in the hands of Mahakasyapa. Know that all that you have learned up to now about the non-eternal and suffering is not true. In spring, for example, people go bathing in a big pond. They are enjoying themselves, sailing in a boat, when they drop a gem of beryl into the depths of the water, after which it can no longer be seen. Then they all get into the water and search for this gem. They competitively scoop up all such rubbish as tiles, stones, bits of wood, and gravel, and say that they have the beryl. They are glad and take the things out, and see that what they hold in their hands is not true. The gem is still in the water. By the power of the gem itself, the water becomes clear and transparent. As a result, the people see that the gemis still in the water, as clearly as when they look up and see the form of the moon in the sky. At that time, there is a wise man there who, working out a power, slowly gets into the water and gains the gem. O you Bhiksus! Do not abide in the thought of the non-Eternal, Suffering, non-Self, and the not-Pure and be in the situation of those people who take stones, bits of wood, and gravel to be the true gem. You must study well the Way, how to act, wherever you go, and “meditate on the Self, the Eternal, Bliss, and thePure”. Know that the outer forms of the four items which you have learnt up to now are inversions and that anyone who desires to practise the Way should act like the wise man who deftly gets hold of thegem. This refers to the so-called thought of Self, and that of the Eternal, Bliss, and Pure."


Now, as far, as I know, Buddhists believe in the concept of "Anatta" i.e. No-Self which seems to contradict the above passage. So how Buddhists reconcile the third chapter of the Mahayana Mahaparinirvana Sutra with the concept of "Anatta"?
I am not an expert on Mahayana Buddhism, but I will try to answer this question.

I found another translation here.

For the same section quoted by OP, this other translation states:


  Bhikṣus, thus you should not cultivate your ideas of impermanence,
  suffering, nonself, impurity, and so forth as if they were the true
  meaning [of the human condition]. Just like those people who took
  rocks, vegetation, or gravel to be the jeweled necklace they were
  looking for, you must carefully learn about expedient means so that
  wherever you are, you can continually cultivate your perception of
  self as well as your perceptions of permanence, bliss, and purity. 


Later on, the text says:


  And out of my desire to subdue the non-Buddhist paths I therefore
  declared: “There is no self,there is no person or [individual] living
  being, life span, personality, observer, actor, or experiencer.” 
  Monks, the heterodox paths affirm a “self” in the same manner as [some
  infer literacy in] the shapes of letters incised into wood
  accidentally by insects. This is why the Tathāgata proclaims “nonself”
  as part of his buddha-dharma. It is because I need to straighten out
  [the thinking] of living beings—because I am aware of their
  situation—that I expound the absence of self. ....
  But what I am speaking of is not what ordinary people imagine the self
  to be. Ordinary people or ignorant people suppose the self to be the
  size of a thumb, or perhaps a mustard seed, or a speck of dust. What
  the Tathāgata explains the self to be is nothing like that. Therefore
  when I preach “dharmas are without self,” in truth they are not
  without self. So what is this self [of which I now speak]? If a dharma
  is true, real, permanent, autonomous, a basis, and its nature is
  immutable, then that is what I call self.


The term "dharma" above means "thing". So "self" is a thing which is true, real, permanent, autonomous, a basis and is immutable.

Then it continues:


  [Kāśyapa continued:] “World-Honored One, is there a self or not in any
  of the twenty-five forms of existence?”
  
  The Buddha said: Good man, “self” is precisely what tathāgatagarbha
  means. All living beings have buddha-nature, and this is what is meant
  by this notion of self. However, the significance of “self” understood
  in this way has been continuously covered over by an uncountable
  number of the defilements since the beginning [of any given
  individual’s existence], and that is why living beings have been
  unable to perceive it.


So, the "self" in the Mahayana Mahaparinirvana Sutra refers to the tathagatha-garbha or Buddha-nature.

It must be noted that tathagatha-garbha or Buddha-nature is not the same as the concept of Atman in Hinduism, as can be seen by the following excerpt from the same text Mahayana Mahaparinirvana Sutra:


  Good man, if people have never heard of the profound, hidden tathā-
  gata treasury, how could they be aware of the existence of
  buddha-nature? What is it that I am calling “a hidden treasury”? I am
  talking about the well-balanced Mahāyāna scriptures themselves. Good
  man, there are other paths: some expound a self that is permanent,
  some expound a self that is cut off permanently [at death]. The
  Tathāgata is not like them. In affirming both self and nonself, I call
  it “the middle path.” One way of explaining this is that the Buddha
  expounds a middle path in which all living beings possess
  buddha-nature, but because it is obscured by the defilements they do
  not understand it and do not see it. Therefore you must diligently
  cultivate whatever expedient means you can in order to destroy those
  defilements.


According to this article:


  Tathagatagarbha, or Tathagata-garbha, means "womb" (garbha) of Buddha
  (Tathagata). This refers to a Mahayana Buddhist doctrine that Buddha
  Nature is within all beings. Because this is so, all beings may
  realize enlightenment. Tathagatagarbha often is described as a seed,
  embryo or potentiality within each individual to be developed.


And it discusses further:


  In the religions of the Buddha's day that were the forerunners of
  today's Hinduism, one of the central beliefs as (and is) the doctrine
  of atman. Atman means "breath" or "spirit," and it refers to a soul or
  individual essence of self. Another is the teaching of Brahman, which
  is understood as something like the absolute reality or the ground of
  being. In the several traditions of Hinduism, the precise relationship
  of atman to Brahman varies, but they could be understood as the small,
  individual self and the big, universal self.
  
  However, the Buddha specifically rejected this teaching. The doctrine
  of anatman, which he articulated many times, is a direct refutation of
  atman.
  
  Through the centuries many have accused the Tathagatagarbha doctrine
  of being an attempt to sneak an atman back into Buddhism by another
  name.
  
  In this case, the potentiality or Buddha-seed within each being is
  compared to atman, and Buddha Nature -- which is sometimes identified
  with the dharmakaya -- is compared to Brahman.
  
  You can find many Buddhist teachers speaking of small mind and big
  mind, or small self and big self. What they mean may not be exactly
  like the atman and Brahman of Vedanta, but it's common for people to
  understand them that way. Understanding Tathagatagarbha this way,
  however, would violate basic Buddhist teaching.


Another article here tries to point the origins of tathagatha-garbha or Buddha-nature to the Luminous Mind in the Pali Canon (which according to Theravada Buddhism, is not permanent or eternal, but is dependently arising - see this answer):


  The term "'tathagatagarbha'" is generally taken as to mean that the
  "garbha" (womb or potential) of a 'Tathagata' exists in all sentient beings without
  exception, and though temporarily contaminated by adventitious
  defilement ('agantukaklesa'), it is the cause which eventually leads
  sentient beings to enlightenment. The notion of the 'tathagatagarbha'
  can be traced to a luminous¡A inherently pure mind (pabhassar citta)
  found in the 'Anguttara-nikaya' (1:5):
  
  Oh! 'Bhiksus'. The mind is pure! It is defiled by the adventitious
  defilement. Oh! 'Bhiksus'. The mind is pure! it obtains liberation
  through the adventitious defilement.
  
  When the original pure mind came to be regarded as something capable
  of growing into Buddhahood, there was the 'tathagatagarbha' doctrine.
  Although the concept of an intrinsically pure mind exists in the
  Nikaya Buddhism, many Buddhologists, such as Wayman (1), Paul (2),
  Yin-shun (3) think that the 'tathagatagarbha' thought was originated
  from the 'Mahasamgika', but was rejected by the 'Theravada'. This
  theory is also held by Mizuno who points out that the pure mind
  ('pabhassarcitta') articulated in the Nikaya Buddhism is not totally
  identical with the original pure mind ('prakrtivisuddhi-citta')
  articulated in the 'Tathagatagarbha' doctrine, for Mizuno asserts that
  the former is static whereas the latter is dynamic in that it is
  capable of eradicating defilement.(4) At any rate, the relationship
  between pure mind and the adventitious defilement appears to have been
  wholly adopted by the 'Mahasamghika' and later by the 'Mahayana'.


It also comments in its conclusion:


  In conclusion, when we try to interpret the thought of the
  'tathagatagarbha', we should keep several points in mind:
  
  1) The 'tathagatagarbha' symbolizes the potential for enlightenment (a
  principle) rather than a material "essence" of ultimate truth,
  
  2) the 'tathagatagarbha' is based on the framework of the 'Mahayana'
  doctrine of 'sunyata-pratitya-samutpada'.
  
  3) The development of the 'tathagatagarbha' doctrine signifies the
  ability of a religious tradition to meet the spiritual needs of the
  masses aiming at a given time.
  
  That is to say the 'tathagatagarbha' thought was formed as an positive
  soterio-logical approach to counteract the "'sunyam sarvam'" (all is
  empty) view. The 'tathagatagarbha' which strongly articulates a
  devotional and experiential approach to salvation provides much to the
  hope and aspiration of the people at large. It is this positive aspect
  that was taken up and strongly emphasized in Chinese Buddhism.
  
  4) The 'tathagatagarbha' doctrine is employed as a skill-in-means
  ('upaya'). This does not necessarily mean that the theory of the
  'tathagatagarbha' is neyartha, a teaching requiring further
  qualifications -- rather, it is a skill-in-means in that it is taught
  to suit the needs of a certain kind of people and circumstances. This
  is why it is said in the 'sutra' that in order to teach the emptiness
  of all dharmas, the Buddhas preach sometimes by the doctrine of the
  'tathagatagarbha', and sometimes by that of emptiness. Thus it is
  better to take the 'tathagatagarbha / Buddha nature' as representing
  "profound existence" derived from "true emptiness" rather than as a
  monistic self.