  Majorana-Weyl spacetimes offer a rich algebraic setup and new types of
space-time dualities besides those discussed by Hull. The triality
automorphisms of Spin(8) act non-trivially on Majorana-Weyl representations and
Majorana-Weyl spacetimes with different signatures. In particular relations
exist among the (1+9)-(5+5)-(9+1) spacetimes, as well as their transverse
coordinates spacetimes (0+8)-(4+4)-(8+0). Larger dimensional spacetimes such as
(2+10)-(6+6)-(10+2) also show dualities induced by triality. A precise
three-languages dictionary is here given. It furnishes the exact translations
among, e.g., the three different versions (one in each signature) of the
ten-dimensional N=1 superstring and superYang-Mills theories. Their dualities
close the six-element permutation group S_3. Bilinear and trilinear invariants
allowing to formulate theories with a manifest space-time symmetry are
constructed.
