Enough with the taus and stuff and should

not be an even function for that? Suppose so then how about just x, and y and u and v and just a real example with T=2 which in that case:

which is a square. Now if I let

and

then that transforms the region in x and y to a diamond shape in the u-v plane going from (2,0) to (0,2) then (-2,0) to (0,-2) and back to (2,0). Then calculate the Jacobian

but the positive direction over the u-v contour goes in the negative direction over the x-y contour so drop the negative sign. Then I get:

assuming

is an even function and is in the form you stated. Looks to me anyway.