The sphere is a perfectly symmetrical object, in that all orientations (unitary transformations of the Cartesian coordinates) look the same.
However, when one tries to parametrize the surface of a sphere, the parameters are non-symmetric. For example, using the standard spherical coordinates, the radial angle $\theta$ acts one way (going from 0 to $2\pi$), and the azimuth angle $\varphi$ acts a different way (going from 0 to $\pi$). You can't interchange these parameters, or apply unitary transforms to them without breaking your parameterization of the sphere's surface.
Is there any other way to parameterize the surface of a sphere (a hyper-sphere in general) such that there's complete symmetry between the parameters?
One way I can think of is to pick any $x$, $y$, and $z$ coordinates and then project them onto the sphere (by dividing by their radius). However, this is a cheat, because 1 of those 3 parameters is essentially no longer free.
P.S.: Another question I have (which could just be a different formulation of the same question) is: is there any parameterization $(u, v)$ of the surface of the sphere such that, for any choice of $(u, v)$, the area $A$ of the rectangle with opposing corners at $(u, v)$ and $(u + du, v + dv)$ will always be the same. For example, this isn't the case with standard spherical coordinates, because rectangles where $\varphi$ is close to $0$ are much smaller than rectangles where $\varphi$ is close to $\frac{\pi}{2}$.