How do I assign functions for cartesian coordinates and rotations to irreducible representations of a character table?

Question

I do not really understand the intuition behind the functions assigned to irreducible representations. For example, in H₂O, the A₂ representation is given by

E   C₂   σv(xz)   σv(yz)
1   -1    1       -1

and the associated linear functions are x and Ry, but I am not sure where these come from.