  We study functions from reals to reals which are uniformly degree-invariant
from Turing-equivalence to many-one equivalence, and compare them "on a cone."
We prove that they are in one-to-one correspondence with the Wadge degrees,
which can be viewed as a refinement of the uniform Martin's conjecture for
uniformly invariant functions from Turing- to Turing-equivalence. Our proof
works in the general case of many-one degrees on $\mathcal{Q}^\omega$ and Wadge
degrees of functions $\omega^\omega\to\mathcal{Q}$ for any better quasi
ordering $\mathcal{Q}$.
