Some ${\mathbf{Z}}_2$-invariant Variables: Constructed of the Scalars Dilaton and Axion

Using the dilaton scalar and axion pseudoscalar fields we construct a number of scalars and differential forms which are symmetric under the ${\mathbf{Z}}_2$-subgroup of the group $SL(2,\mathbf{R})$. These invariants enable us to establish various 10-dimensional invariant actions. Other invariants which are not independent from the previous ones will be detached.