  Twistor string models have been known for more than a decade now but have
come back under the spotlight recently with the advent of the scattering
equation formalism which has greatly generalized the scope of these models. A
striking ubiquitous feature of these models has always been that, contrary to
usual string theory, they do not admit vibrational modes and thus describe only
conventional field theory. In this paper we report on the surprising discovery
of a whole new sector of one of these theories which we call "twisted strings,"
when spacetime has compact directions. We find that the spectrum is enhanced
from a finite number of states to an infinite number of interacting higher spin
massive states. We describe both bosonic and world sheet supersymmetric models,
their spectra and scattering amplitudes. These models have distinctive features
of both string and field theory, for example they are invariant under stringy
T-duality but have the high energy behavior typical of field theory. Therefore
they describe a new kind of field theories in target space, sitting on their
own halfway between string and field theory.
