  We show that Ramanujan-type congruences are preserved by the action of the
shallow Hecke algebra and provide several structure results for them. We
discover a dichotomy between congruences originating in Hecke eigenvalues and
congruences on arithmetic progressions with cube-free periods. The scarcity of
the latter was investigated recently. We explain that they provide congruences
among algebraic parts of twisted central $\mathrm{L}$-values. We specialize our
results to partition congruences, for which we investigate the proofs of
partition congruences by Atkin and by Ono, and develop a heuristic that
suggests that their approach by Hecke operators acting diagonally modulo $\ell$
on modular forms is optimal. In an extended example, we showcase how to employ
our conclusions to benefit experimental work.
