  Bimetric MOND (BIMOND) theories, propounded recently, predict peculiar
gravitational interactions between matter and twin matter (TM). Twin matter is
hypothetical matter that might couple directly only to the second metric of the
theory. Considerations of cosmology in the BIMOND framework suggest that such
TM might exist and copy matter in its attributes. Here I investigate the
indirect interactions that BIMOND theories predict between nonrelativistic
masses of matter and TM. The most salient result is that in the deep-MOND
regime of the matter-TM-symmetric theories, TM behaves as if it has a negative
gravitational mass. To wit, interaction within each sector is attractive MOND
gravity, but between matter and TM it is repulsive MOND gravity. In the
high-acceleration regime, the interaction depends on a parameter, beta. For the
favored value beta=1, matter and TM do not interact in this regime; for beta<1
they attract; and for beta>1 they repel each other. Such interactions may have
substantial ramifications for all aspects of structure formation, such as
matter distribution, peculiar velocities, and effects on the CMB. The repulsive
interactions probably lead to segregation of matter and TM structures, leading,
in turn, to intermeshing of the respective cosmic webs, with high-density nodes
of one sector residing in the voids of the other (possibly conducing to
efficient evacuation of the voids). Weak lensing by TM seems the best way to
constrain its attributes. In the MOND regime a TM body acts on matter photons
as a diverging lens. Strong lensing occurs in the high acceleration regime, and
thus depends on the value of beta. For beta=1 no strong lensing effects of TM
are expected. I also discuss briefly asymmetric theories (abridged).
