  Motivated by the conjectures formulated in 2003 by Tun\c{c}el et al., we
study interlacing properties of the eigenvalues of $A\otimes B + B\otimes A$
for pairs of $n$-by-$n$ matrices $A, B$. We prove that for every pair of
symmetric matrices (and skew-symmetric matrices) with one of them at most rank
two, the \emph{odd spectrum} (those eigenvalues determined by skew-symmetric
eigenvectors) of $A\otimes B + B\otimes A$ interlaces its \emph{even spectrum}
(those eigenvalues determined by symmetric eigenvectors). Using this result, we
also show that when $n \leq 3$, the odd spectrum of $A\otimes B + B\otimes A$
interlaces its even spectrum for every pair $A, B$. The interlacing results
also specify the structure of the eigenvectors corresponding to the extreme
eigenvalues. In addition, we identify where the conjecture(s) and some
interlacing properties hold for a number of structured matrices. We settle the
conjectures of Tun\c{c}el et al. and show they fail for some pairs of symmetric
matrices $A, B$, when $n\geq 4$ and the ranks of $A$ and $B$ are at least $3$.
