Let p be a prime greater than 3, X = { 1, 2, ..., p-1 } and R the set of primitive roots mod p contained in X. To each g ∈ R associate the permutation σ_g of X defined by σ_g(x) = y where y is the unique member of X satisfying y ≡ g^x mod p. Let Σ_R = { σ_g | g ∈ R }. We analyze the parity of the permutations in Σ_R. If p ≡ 1 mod 4 half the permutations are even and half are odd. If p ≡ 3 mod 4 they are either all even or all odd; set ε(p) = 1 in the even case, ε(p) = -1 in the odd case. Numerical evidence suggests the conjecture that ε(p) ≡ h(-p) mod 4, where h(-p) is the class number of the quadratic field Q(\sqrt{-p}). The conjecture is shown to be true, and furthermore ε(p) ≡ -((p-1/2))! mod p. We also study a larger class of permutations of degree p-1 which generalize the Σ_R.