  Split-step quantum walk operators can be expressed as a generalised version
of CMV operators with complex transmission coefficients, which we call rotated
CMV operators. Following the idea of Cantero, Moral and Velazquez's original
construction of the original CMV operators from the theory of orthogonal
polynomials on the unit circle (OPUC), we show that rotated CMV operators can
be constructed similarly via a rotated version of OPUCs with respect to the
same measure, and admit an analogous LM-factorisation as the original CMV
operators. We also develop the rotated second kind polynomials corresponding to
the rotated OPUCs. We then use the LM-factorisation of rotated alternate CMV
operators to compute the Gesztesy-Zinchenko transfer matrices for rotated CMV
operators.
