  A contragenic function in a domain $\Omega\subseteq\mathbf{R}^3$ is a
reduced-quaternion-valued (i.e. the last coordinate function is zero) harmonic
function, which is orthogonal in $L^2(\Omega)$ to all monogenic functions and
their conjugates. The notion of contragenicity depends on the domain and thus
is not a local property, in contrast to harmonicity and monogenicity. For
spheroidal domains of arbitrary eccentricity, we relate standard orthogonal
bases of harmonic and contragenic functions for one domain to another via
computational formulas. This permits us to show that there exist nontrivial
contragenic functions common to the spheroids of all eccentricities.
