  We analyze N = 1 theories on S5 and S4 x S1, showing how their partition
functions can be written in terms of a set of fundamental 5d holomorphic
blocks. We demonstrate that, when the 5d mass parameters are analytically
continued to suitable values, the S5 and S4 x S1 partition functions degenerate
to those for S3 and S2 x S1. We explain this mechanism via the recently
proposed correspondence between 5d partition functions and correlators with
underlying q-Virasoro symmetry. From the q-Virasoro 3-point functions, we
axiomatically derive a set of associated reflection coefficients, and show they
can be geometrically interpreted in terms of Harish-Chandra c-functions for
quantum symmetric spaces. We then link these particular c-functions to the
types appearing in the Jost functions encoding the asymptotics of the
scattering in integrable spin chains, obtained taking different limits of the
XYZ model to XXZ-type.
