  Cloaking is typically reciprocal. We introduce here the concept of
\emph{transmittable nonreciprocal cloaking} whereby the cloaking system
operates as a standard omnidirectional cloak for external illumination, but can
transmit light from its center outwards at will. We demonstrate a specific
implementation of such cloaking that consists in a set of concentric
bianisotropic metasurfaces whose innermost element is nonreciprocal and
designed to simultaneously block inward waves and pass -- either
omnidirectionaly or directionally -- outward waves. Such cloaking represents a
fundamental diversification of conventional cloaking and may find applications
in areas such as stealth, blockage avoidance, illusion and cooling.
