When does the ‘iddah in the case of revocable divorce end? Is it when the menstrual bleeding ceases and the woman sees the tuhr (white discharge signalling the end of menses), or is it essential for her to do ghusl?
Question

I divorced my wife on 20/9/2012, and two days after that she got her menses, and she completed the third menses on 9/11/2012, i.e., after approximately fifty days. It was my intention to take her back, but she asked me to tell her before I took her back, and I did not expect three menstrual cycles to pass so quickly. Hence when I found out, I took her back before she purified herself following the third menses. But she told me that the bleeding had ceased but she had not purified herself, out of laziness on her part, and she did not agree to come back to me, as she says that her ‘iddah had ended. That is because she wants to come back subject to certain conditions. Is she now my wife on the grounds that I took her back, or has she become a stranger to me, such that it is not permissible for me to look at her as a wife until after a new marriage contract and a new mahr? Please note that I have not confirmed the divorce in any official way.
Praise be to Allah.Firstly: 
The basic principle is that a woman’s word is to be accepted with regard to the end of the ‘iddah, if she claims that within a timeframe in which it is possible that the ‘iddah could have ended. Ibn Qudaamah (may Allah have mercy on him) said: If a woman claims that her ‘iddah has ended, within a timeframe in which it is possible for it to have ended, then her word is to be accepted, because Allah, may He be exalted, says (interpretation of the meaning): “and it is not lawful for them to conceal what Allah has created in their wombs” [al-Baqarah 2:228]. It was said in the tafseer that this refers to menses and pregnancy. Were it not for the fact that their word is acceptable, they would not have been instructed not to conceal it. Moreover, because this is a matter that only women would know, what counts is the woman’s own word concerning it. 
End quote from al-Mughni (7/406). See also: ash-Sharh al-Mumti‘ (13/198) 
Secondly: 
Once it is established that what counts is the woman’s word with regard to the end of the ‘iddah – as explained above – and we accept your wife’s claims that the bleeding had ended, then in this case, taking her back occurred after the menses had ended and before she did ghusl. With regard to taking the wife back in that case, there is a difference of scholarly opinion as to whether it is valid or not. There are two views: 
The first view is that taking her back is valid so long as the woman has not yet done ghusl following the third menses. 
Those scholars who hold this view quoted as evidence for it the reports narrated from a number of the Sahaabah, including the following: 
The report narrated by ‘Abd ar-Razzaaq in al-Musannaf and al-Haythami in Majma‘ az-Zawaa’id, from ‘Abdullah ibn Mas‘ood (may Allah be pleased with him), according to which he was with ‘Umar ibn al-Khattaab (may Allah be pleased with him) when a man and his wife came to him and the man said: I divorced my wife, then I took her back. The woman said: Because of your treatment, I did not tell you exactly what happened. ‘Umar (may Allah be pleased with him) said: Say what exactly happened. She said: He divorced me then he left me until  three menstrual cycles had passed and the bleeding had ceased, and I prepared my water for ghusl and closed my door and took off my garment, then he knocked on the door and said: I have taken you back, I have taken you back. So I forgot about ghusl and put on my clothes. ‘Umar (may Allah be pleased with him) said: What do you say about it, O Ibn Umm ‘Abd [‘Abdullah ibn Mas‘ood]? I said: I think he has more right to take her back so long as prayer has not become permissible for her (i.e., so long as she has not yet done ghusl). ‘Umar said: That is a good opinion and I think the same.
It was narrated by ‘Abd ar-Razzaaq in al-Musannaf and by al-Haythami in Majma‘ az-Zawaa’id. 
Al-Haythami (may Allah have mercy on him) said: Its men are the men of as-Saheeh. End quote from Majma‘ az-Zawaa’id (4/621) 
It was also narrated by ‘Abd ar-Razzaaq in his Musannaf that ‘Ali said concerning a man who divorced his wife once or twice: It is permissible for her husband to take her back, until she does ghusl following the third menses and prayer becomes permissible for her. 
The second view is that taking her back is not valid after the bleeding ceases in the third menses and the tuhr (white discharge signalling the end of menses) is seen, even if she has not yet done ghusl. 
Those who hold this view quoted as evidence the verse in which Allah, may He be exalted, says (interpretation of the meaning): “And their husbands have the better right to take them back in that period” [al-Baqarah 2:228]. They said: The phrase “in that period” refers to the ‘iddah or waiting-period mentioned, which is three menstrual cycles. So the ‘iddah is ordained as the timeframe within which the husband can take the wife back. What this verse implies is that if these menstrual cycles end without him having taken her back, then he does not have the right to take her back (after that). 
Ibn Qudaamah (may Allah have mercy on him) said: If the woman’s menses ended for the third time, and she has not done ghusl, does her ‘iddah come to an end when she sees the tuhr (white discharge signalling the end of menses)? There are two views concerning this which were narrated by Ibn Haamid. 
1.The first view is that ‘iddah does not end until she does ghusl, and her husband may take her back during that time… This is the view of many of our companions, and that was narrated from ‘Umar, ‘Ali and Ibn Mas‘ood. … And it is not known that anyone differed with them during their period, so there was consensus on this point. Moreover, most of the rulings on menses do not cease to apply except after ghusl is done, and this principle also applies here.
2.The second view is that the ‘iddah ends as soon as the tuhr (white discharge signalling the end of menses) is seen, before doing ghusl … because Allah, may He be exalted, says (interpretation of the meaning): “And divorced women shall wait (as regards their marriage) for three menstrual periods” [al-Baqarah 2:228]. The menstrual period is the menses, which has ended, so the waiting has also ended. End quote from al-Mughni (7/402-403) 
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said, commenting on az-Zaad:
“If she has seen the tuhr (white discharge signalling the end of menses) following the third menses and has not yet done ghusl, then he may take her back.” There are two scholarly views concerning this issue; it is one of the major issues concerning which the evidence for the two views is almost equal in strength. 
Then he (may Allah have mercy on him) quoted the two views concerning the issue, but he inclined towards the view that it is valid for the man to take his wife back so long as she has not yet done ghusl following her menses, quoting as evidence the reports narrated from the Sahaabah. End quote from ash-Sharh al-Mumti‘ (13/193-194) 
Conclusion: 
This issue is one concerning which there is a difference of scholarly opinion, and there is no clear text concerning it. Even if we go by the view that it is valid to take back the wife in that situation, what appears to be the case with regard to the wife whom you have divorced is that she does not want to go back to you because she believes that her ‘iddah has ended. So as to avoid disputes, we advise you to refer to the qaadi (Islamic judge) so that he may decide concerning the matter. 
It should be noted that if the husband issues a divorce, it counts as such even if he does not document that divorce in the courts. For more information, please see the answer to question no. 169624
And Allah knows best.