What was the basis for saying that 11th-century Greek churches were "rebaptizing" Latins? In the run-up to the fateful excommunications of 1054, accusatory letters were exchanged between the Patriarch of Constantinople, Michael Cerularius, and Pope Leo IX.  Each had issues with the other's practices, but one in particular mentioned by Leo IX stands out to me – the "rebaptism of Latins by Greeks," in the words of Everett Ferguson (Church History, I, 19.V.A).

This claim strikes me as odd, because I don't believe that that's the current practice of the Eastern Orthodox.  Were the Greeks actually "rebaptizing" Latins at this point in history?  If so, why?  And if not, where did Leo IX get this idea?
Some Greeks were rebaptizing Latins. While this does not seem to have been the universal position of the Orthodox Church in XI Century, some of the Eastern clergy objected to the Latin method of baptism. In particular, it was believed in the East that the West baptized by single immersion. The East used triple immersion. Some in the East appealed to Canon 7 of the First Council of Constantinople:


  But Eunomians, who are baptized with only one immersion, ... all these, when they desire to turn to orthodoxy, we receive as heathen ... and then we baptize them.


Note that Canon 7 diffentiates some heretical groups to be reconciled through Confession and Chrismation:


  Arians, and Macedonians, and Sabbatians, [more groups], we receive, upon their giving a written renunciation and anathematize every heresy which is not in accordance with the Holy, Catholic, and Apostolic Church of God. Thereupon, they are first sealed or anointed with the holy oil upon the forehead, eyes, nostrils, mouth, and ears; and when we seal them, we say, “The Seal of the gift of the Holy Ghost.”


So it seems that the canon was concerned with the method of baptism in addition to heretical beliefs. Canon 95 of the Quinisext Council in Trullo is similar. Whether this canon was appropriately applied is debatable, but it was used as an excuse.

Some other Greeks objected to that Latin formula "I baptized you in the Name...." as it was perceived to emphasize the action of the priest. They preferred the formula "The servent of God is baptized in the Name...." as it was seen to emphasize the work of God. I don't know if they asserted canonical support for this argument, but it caused some Greeks to question the validity of the Latin method. 

The filioque does not seem to be directly involved here. While the Greeks considered it to be heretical, rebaptism would not be required. Canonically, a heretic is reconciled by Confession and Chrismation.