  Originating from spectral graph theory, cospectrality is a powerful
generalization of exchange symmetry and can be applied to all real-valued
symmetric matrices. Two vertices of an undirected graph with real edge weights
are cospectral iff the underlying weighted adjacency matrix $M$ fulfills
$[M^k]_{u,u} = [M^k]_{v,v}$ for all non-negative integer $k$, and as a result
any eigenvector $\phi$ of $M$ has (or, in the presence of degeneracies, can be
chosen to have) definite parity on $u$ and $v$. We here show that the powers of
a matrix with cospectral vertices induce further local relations on its
eigenvectors, and also can be used to design cospectrality preserving
modifications. To this end, we introduce the concept of \emph{walk equivalence}
of cospectral vertices with respect to \emph{walk multiplets} which are special
vertex subsets of a graph. Walk multiplets allow for systematic and flexible
modifications of a graph with a given cospectral pair while preserving this
cospectrality. The set of modifications includes the addition and removal of
both vertices and edges, such that the underlying topology of the graph can be
altered. In particular, we prove that any new vertex connected to a walk
multiplet by suitable connection weights becomes a so-called unrestricted
substitution point (USP), meaning that any arbitrary graph may be connected to
it without breaking cospectrality. Also, suitable interconnections between walk
multiplets within a graph are shown to preserve the associated cospectrality.
Importantly, we demonstrate that the walk equivalence of cospectral vertices
$u,v$ imposes a local structure on every eigenvector $\phi$ obeying $\phi_{u} =
\pm \phi_{v} \ne 0$ (in the case of degeneracies, a specific choice of the
eigenvector basis is needed). Our work paves the way for flexibly exploiting
hidden structural symmetries in the design of generic complex network-like
systems.
