  Two vertices $a$ and $b$ in a graph $X$ are cospectral if the vertex-deleted
subgraphs $X\setminus a$ and $X\setminus b$ have the same characteristic
polynomial. In this paper we investigate a strengthening of this relation on
vertices, that arises in investigations of continuous quantum walks. Suppose
the vectors $e_a$ for $a$ in $V(X)$ are the standard basis for
$\mathbb{R}^{V(X)}$. We say that $a$ and $b$ are strongly cospectral if, for
each eigenspace $U$ of $A(X)$, the orthogonal projections of $e_a$ and $e_b$
are either equal or differ only in sign. We develop the basic theory of this
concept and provide constructions of graphs with pairs of strongly cospectral
vertices. Given a continuous quantum walk on on a graph, each vertex determines
a curve in complex projective space. We derive results that show tht the closer
these curves are, the more "similar" the corresponding vertices are.
