Or do they always keep white on the bottom, as is the case with most historic games? Wouldn't it be easier for them to spot patterns from old games if they always oriented the board the same way in their minds?
In Blindfold Chess (Hearst-Knott; reviewed and summarized here), a natural beginning place for studying the thought processes of chess masters, the authors discover this (quote from the link):
blindfold chess masters consistently report that what they visualize are not images of pieces or chessboards, but abstractions of these with minimal or no physical features.
And later, they describe their mental state in a variety of telling ways:
“no mental pictures,” “abstract knowledge,” “I know where the pieces are,” “only an abstract type of representation,” “only relationships,” “no real picture,” “the significance of a piece,” “knowing what combination or plan is in progress,” “lines of force,” “pieces are only friend or foe, carriers of particular actions,” “sort of formless visions of the positions,” and so forth. Many of the masters report that they have no mental image at all (p.151).
This suggests that your question is actually based on a faulty premise: that chess masters explicitly visualize a board while contemplating a position. Instead of colored pieces "visualized" on a board, they seem likely to think about the position in a more abstract fashion.
The source covers blindfold chess, naturally, but I would be astonished if the same concepts were not true for most masters in general.
I'm not master level myself, but for what it's worth, I would also describe my mentalization process in similar, abstract terms. I almost never "picture" the board.