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What does the 2 in t2g stand for?

I have read so far that it is about whether the d-Orbital is symmetric to a C2 element perpendicular to its main rotational axis. If all the given orbitals in a group are symmetric to that element, you assign a 1 towards it, otherwise a 2. Anyway, I still don't get a few things:

First: Is it enough for one orbital of a given group to be not symmetric concerning that C2 element, for the whole group being asymmetric towards C2?

Second: I cannot get my head around e1 and e2 in the crystal field splitting of linear complexes. Do I have to keep the C2 axis the same to check whether it is e1 or e2? Because if I change the axis I get e1 twice, if I don't change it, I get e2 twice. My textbook says it is e1 for d(xz)/d(yz) and e2 for d(x2-y2)/d(xy)...