  Two emerging topics in graph theory are the study of cospectral vertices of a
graph, and the study of isospectral reductions of graphs. In this paper, we
prove a fundamental relationship between these two areas, which is that two
vertices of a graph are cospectral if and only if the isospectral reduction
over these vertices has a nontrivial automorphism. It is well known that if two
vertices of a graph are symmetric, i.e. if there exists a graph automorphism
permuting these two vertices, then they are cospectral. This paper extends this
result showing that any two cospectral vertices are symmetric in some reduced
version of the graph. We also prove that two vertices are strongly cospectral
if and only if they are cospectral and the isospectral reduction over these two
vertices has simple eigenvalues. We further describe how these results can be
used to construct new families of graphs with cospectral vertices.
