  We study globally defined $(\lambda,\mu)$-eigenfamilies on compact Riemannian
manifolds. Among others, we provide (non-) existence results for such
eigenfamilies, examine their topological properties and classify
$(\lambda,\mu)$-eigenfamilies on flat tori. It is further shown that for
$f=f_1+i f_2$ being an eigenfunction decomposed into its real and its imaginary
part, the powers $\{f_1^a f_2^b\mid a,b\in\mathbb N\}$ satisfy highly rigid
orthogonality relations in $L^2(M)$. In establishing these orthogonality
relations one is led to combinatorial identities involving determinants of
products of binomials, which we view as being of independent interest.
