  We describe the properties of evanescent gravitational waves (EGWs)---wave
solutions of Einstein equations which decay exponentially in some direction
while propagating in another. Evanescent waves are well-known in acoustics and
optics and have recently received much attention due to their extraordinary
properties such as their transverse spin and spin-momentum locking. We show
that EGWs possess similarly remarkable properties, carrying transverse spin
angular momenta and driving freely falling test masses along in elliptical
trajectories. Hence, test masses on a plane transverse to the direction of
propagation exhibit correlated vector and scalar-like
deformation---correlations which can be used to distinguish it from modified
gravity. We demonstrate that EGWs are present and dominant in the vicinity of
sub-wavelength sources such as orbiting binaries.
