Mind you that this is, of course, a roleplaying issue only and not a mechanical one. But if you're playing anything more than a kick-in-the-door, beer-and-pretzel game, then how a cleric might use an incompatible holy symbol is a question that shouldn't even come up, because no one should even suggest the idea.
I think I disagree. We're not just talking about symbols of other gods, we're talking about powerful magical items that the cleric can use to actually further his/her god's agenda.
Refusing to actually accomplish something because you'd have to use an item with another ideology's mark on it seems like it might actually annoy your god in some situations.
Of course the cleric should wield a symbol of his/her own god (and possibly several mundane ones), but if there's a good reason to also have another one (remember, clerics only need to wear the symbol, not hold it out or anything) he/she should probably at least consider it.