Dualization of Higher Derivative Heterotic Supergravities in $6D$ and $10D$

There exist two four-derivative extensions of $N=(1,0)$ supergravity in six dimensions. A particular combination of them is known to dualize to the analog of the the Bergshoeff-de Roo (BdR) action in $10D$. Here we first show that the two extensions are not related to each other by any field redefinitions. Next, we dualize them separately thereby obtaining a two parameter dual theory. This is done directly at the level of the action, thus avoiding the laborious method of integrating equations of motion of the dualized theory into an action. To explore whether a similar phenomenon exists in $10D$, we study the dualization of the BdR action in $10D$ in detail. We find an obstacle in the separation of the result into a sum of two independent invariants because of the presence of terms which do not lift from $6D$ to $10D$. We also compare the dual of the BdR action with an existing result obtained in superspace. We find that the bosonic actions agree modulo field redefinitions.