  We present component and superspace formulations for the recently-proposed
Type IIA* (or so-called `star') supergravity theory, which is timelike dual to
the conventional Type IIB theory. First, within the component approach, all
terms in the action are fixed up to the quartic fermionic ones. As desired, the
kinetic terms for Ramond-Ramond fields have signs opposite to the conventional
case. Consistency of these are then insured by the construction of a superspace
description of this theory. As a by-product, we find that a single signature
parameter $s = \pm 1$ can interpolate the Type IIA and Type IIA* theories in
superspace. This superspace result naturally allows us to present a
Green-Schwarz action, that possesses $\kappa$-symmetry, consistent with such
backgrounds. We also give general algebraic descriptions of such `star'
theories, so that they can be identified as representatives of some of the
equivalence classes of $\kappa$-invariant Green-Schwarz actions.
