We say that a matrix R∈Cn×n is k-involutory if its minimal polynomial is xk-1 for some k⩾2, so Rk-1=R-1 and the eigenvalues of R are 1, ζ, ζ2,…,ζk-1, where ζ=e2πi/k. Let α,μ∈{0,1,…,k-1}. If R∈Cm×m,A∈Cm×n,S∈Cn×n and R and S are k-involutory, we say that A is (R,S,α,μ)-symmetric if RAS-α=ζμA. We show that an (R,S,α,μ)-symmetric matrix A can be usefully represented in terms of matrices Fℓ∈Ccαℓ+μ×dℓ,0⩽ℓ⩽k-1, where cℓ and dℓ are respectively the dimensions of the ζℓ-eigenspaces of R and S. This continues a theme initiated in an earlier paper with the same title, in which we assumed that α=1. We say that a k-involution is equidimensional with width d if all of its eigenspaces have dimension d. We show that if R and S are equidimensional k-involutions with widths d1 and d2 respectively, then (R,S,α,μ)-symmetric matrices are closely related to generalized α-circulants [ζμrAs-αr]r,s=0k-1, where A0,A1,…,Ak-1∈Cd1×d2. For this case our results are new even if α=1. We also give an explicit formula for the Moore-Penrose inverse of a unilevel block circulant [As-αr]r,s=0k-1 for any α∈{0,1,…,k-1}, generalizing a result previously obtained for the case where gcd(α,k)=1.