How to morph a 2d grid of saturation and luminance onto the surface of a torus?

Question

For any given hue, we get a Cartesian grid of saturation and luminance like so:

I would like to warp this surface to the shape of a torus:

such that all the colors are continuous, and that there is a one-to-one correspondence between a point on the grid to a point on the surface of the torus. By continuous colors I mean that every path on the torus is essentially a smooth gradient of colors. Can this be done?

Answer 1

Yes if you are willing to allow some duplication.

Reflect that square (call it A) over its right and top edges to create two new squares B and C so the three form an ell shape

B
A C

Then reflect B or C to make D.

B D
A C

Finally, glue the horizontal and vertical edges of the new big square in the usual way to make a torus.