The orbitals are real physically, but the interesting thing is that they are derived/predicted from the mathematics of the wave equation.
There are 5 d orbitals so you would think that there are five solution to the equation. As it turns out , the math actually produces six solutions to Schroedinger equation for l = 2, but there can only be five linearly independent solutions. The standard procedure in differential calculus is to use a linear combination of two dependent functions to produce one independent one. Thus, dz2 looks odd because it is a linear combination of d(z2 - x2) and d(z2 - y2) functions.