  In quantum scattering, Hermiticity is necessary for both reciprocity and
unitarity. Reciprocity means that both reflectivity (R) and transmitivity (T)
are insensitive to the direction of incidence of a wave (particle) at a
scatterer from left/right. Unitarity means that R+T=1. In scattering from
non-Hermitian PT-symmetric structures the (left/right) handedness
(non-reciprocity) of reflectivity is known to be essential and unitarity
remains elusive so far. Here we present a surprising occurrence of both
reciprocity and unitarity in scattering from a complex PT-symmetric potential.
In special cases, we show that this potential can even become invisible (R=0,
T=1) from both left and right sides. We also find that this potential in a
parametric regime enjoys a pseudo-unitarity of the type
$T+\sqrt{R_{left}R_{right}}=1$.
