  We explore the scattering of waves in designed asymmetric one-dimensional
waveguide networks. We show that the reflection between two ports of an
asymmetric network can be identical over a broad frequency range, as if the
network was mirror-symmetric, under the condition of so-called latent symmetry
between the ports. This broadband equireflectionality is validated numerically
for acoustic waveguides and experimentally through measurements on microwave
transmission-line networks. In addition, introducing a generalization of latent
symmetry, we study the properties of an $N$-port scattering matrix $S$. When
the powers of $S$ fulfill certain relations, which we coin scaled
cospectrality, the setup is guaranteed to possess at least one zero eigenvalue
of $S$, so that the setup features coherent perfect absorption. More
importantly, scaled cospectrality introduces a scaling factor which controls
the asymmetry of the incoming wave to be absorbed. Our findings introduce a
novel approach for designing tunable wave manipulation devices in asymmetric
setups. As evidenced by our acoustic simulations and microwave experiments, the
generality of our approach extends its potential applications to a wide range
of physical systems.
