What decides membrane mode shape 'direction'? Consider the modes of a circular membrane which has perfectly isotropic mechanical properties. The fundamental mode 01, has a symmetric shape as does 02 etc. However, for example the mode 11 has a direction. i.e. there is a line along which the max peak and trough fall like a sine function. 

For a 1D element like a string, the direction is always fixed so there is no ambiguity. What decides this direction for a 2D membrane? Is there some constraint beyond the Youngs modulus (e.g. Poissons ratio). Or is it an experimental thing (there can never be a perfectly circular membrane)?
 A: If you do the math, you find that modes like 11 and 21 actually occur in pairs. There are two identical eigenvalues (i.e. natural frequencies), and therefore the eigenvectors are not unique. The eigenvectors can be any linear combination of two "basic eigenvectors". For the 11 and 21 modes the basic eigenvectors are identical to each other (and as drawn in your picture) except that their nodal diameters (where the displacement is always zero) are perpendicular to each other.
By taking linear combinations of those two shapes, with a phase difference between them, you can get the "same" shape with any orientation of the nodal diameters.
So in theory, the plate can vibrate at its natural frequency with the nodal diameters in any arbitrary direction.
In real life, as you suggested, the plate and its boundary conditions will never be perfectly symmetrical, and the small imperfections will "fix" the nodal diameters in some definite orientation.
In fact, every natural frequency of the plate will have a pair of eigenvectors, except for the special case when the first index of the mode is $0$. Mathematically, this is analogous to the fact that a Fourier series only has one constant term, but all the other terms in the series occur in pairs - i.e. the corresponding $\sin$ and $\cos$ terms in the series. (Of course the reason there is only one constant term is because $\sin 0x = 0$ for all values of $x$).
