What is the meaning of the shapes of various orbitals of the atom? [duplicate]

I understand that the orbitals are a 3D contour inside of which there is a certain probability ( for example 90% ) to find the electron ( this is my understanding of it, so correct me if I am wrong ).

Now, I want to know the meaning of shapes like that of p-orbital. They are dumbbell-shaped ( sort of ) and are located on the x-, y- and z-axis. But aren't the axis completely arbitrary. Since the axis are arbitrary, we should expect only shapes that remain invariant upon any kind of rotations of the axis. The only thing that fits in that criterion is the spherical shape (or s-orbital).

So, what is the meaning of the non-spherical shapes of various orbitals, in the context I have mentioned above?

marked as duplicate by Emilio Pisanty quantum-mechanics StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jul 30 '17 at 13:48

However, we have chosen axes to describe the system and all our results are given with respect to these axes. Further, we arbitrarily chose a full set of commuting operators, whose eigenvalues uniquely specify any eigenvector of the hamiltonian. These operators are usually taken to be $H, L^2, L_z$ with their eigenvalues labled by $n$, $l$, $m$ respectively. This makes the $z$-axis special to our description of the system, while this of course it is still just an artifact of our choice of coordinates.