  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.
