Does MN9 imply a cyclic causation? How does it work? The Nanamoli and Bodhi translation of MN9 states in 2 excerpts:


  With the arising of the taints there is the arising of ignorance. With
  the cessation of the taints there is the cessation of ignorance. .....
  
  With the arising of ignorance there is the arising of the taints.
  With the cessation of ignorance there is the cessation of the taints.


The same 2 excerpts from the Thanissaro translation of MN9 reads:


  From the origination of fermentation comes the origination of
  ignorance. From the cessation of fermentation comes the cessation of
  ignorance. .....
  
  From the origination of ignorance comes the origination of
  fermentation. From the cessation of ignorance comes the cessation of
  fermentation.


When I took a look at the Pali version of Sammaditthi Sutta (MN9), I'm guessing that these phrases refer to:


  Āsavasamudayā avijjāsamudayo, āsavanirodhā avijjānirodho .....
  
  Avijjāsamudayā āsavasamudayo, avijjānirodhā āsavanirodho


How does this logic work?

When A arises, B arises. When A ceases, B ceases.

When B arises, A arises. When B ceases, A ceases.

Does this imply a cyclic causation (similar to chicken and egg)? Why is this the case? How does it work? How can the cycle be broken?
Yup, it's a vicious circle. On one side we have Ignorance/Confusion/Delusion and on the other Stress, Negative Emotions, and Bad Karma.

It is like being born in a poor uneducated family: because you are poor, you can't get education, and because you can't get education you are poor.

When you are stressed, you don't have time to stop and seek solution, you keep rolling day after day, never acting as per your plan, but only reacting to your issues. Because you keep rolling uncontrollably, very stressed, your head is not clear, your will is damaged, your perspective is skewed, so you keep sowing seeds for future trouble.

This is called "bewilderment", or as my teacher said:


  One misstep begets another, begets another, and on, and on.


Imagine someone walking on a trail of rocks in the middle of a swamp, carrying a heavy backpack. Once you lose your balance and misstep off a rock, you're at a danger of falling in the swamp, so you have to compensate by making another sharp step sideways, but it does not end there - the inertia of the heavy backpack keeps carrying you on, so you have to keep making these sharp steps. You are lucky if you can stabilize yourself, but very often you end up running out of rocks and falling in the swamp.

The opposite of this, as often portrayed in suttas, is the good cycle of Sila/Samatha/Vipashyana. The more you fix your karma, the calmer you get, the more clearly you see, which allows you to further optimize your physical and mental action, which leads to even more peace, and even more clarity of vision and so on.