  We establish a strong categorical correspondence between isomorphism classes
of sheaves of arbitrary rank on one algebraic curve and twisted pairs on
another algebraic curve. In a particular application, we realize a generic
elliptic curve as a spectral cover of the complex projective line
$\mathbb{P}^1$ and then construct examples of semistable co-Higgs bundles over
$\mathbb{P}^1$ as pushforwards of locally-free sheaves of certain small ranks
over the elliptic curve. By appealing to a composite push-pull projection
formula, we conjecture an iterated version of the spectral correspondence. We
prove this conjecture for a particular class of spectral covers of $\mathbb
{P}^1$. The proof relies upon a classification of Galois groups into primitive
and imprimitive types. In this context, we revisit a century-old theorem of
J.F. Ritt.
