Automorphisms of toric varieties and Gale duality

We classify complete toric threefolds $X$ such that the automorphism group $\text{Aut}(X)$ acts on $X$ with an open orbit whose complement does not contain a divisor. The latter condition means that for any ray $\rho$ of the fan ${\mathrm{\Sigma }}_X$ there is a Demazure root of ${\mathrm{\Sigma }}_X$ associated with $\rho$. We also find among these varieties those $X$ for which the group $\text{Aut}(X)$ is transitive on the smooth locus $X^{\text{reg}}$. The classifications are based on Gale-dual interpretations of these properties.