  We study the spectral properties of flipped Toeplitz matrices of the form
$H_n(f)=Y_nT_n(f)$, where $T_n(f)$ is the $n\times n$ Toeplitz matrix generated
by the function $f$ and $Y_n$ is the $n\times n$ exchange (or flip) matrix
having $1$ on the main anti-diagonal and $0$ elsewhere. In particular, under
suitable assumptions on $f$, we establish an alternating sign relationship
between the eigenvalues of $H_n(f)$, the eigenvalues of $T_n(f)$, and the
quasi-uniform samples of $f$. Moreover, after fine-tuning a few known theorems
on Toeplitz matrices, we use them to provide localization results for the
eigenvalues of $H_n(f)$. Our study is motivated by the convergence analysis of
the minimal residual (MINRES) method for the solution of real non-symmetric
Toeplitz linear systems of the form $T_n(f)\mathbf x=\mathbf b$ after
pre-multiplication of both sides by $Y_n$, as suggested by Pestana and Wathen.
