Over counting with symmetry

A Senate committee has $5$ Democrats and $5$ Republicans. In how many ways can they seat around a circular table if all the members of each party sit next to each other? Two seating arrangements are considered the same if, for each person, to his or her left is the same in both arrangements.

My Approach:

We can arrange each group in $5!$ ways. We can arrange the group as whole in $2!$ ways i.e. $RD$ and $DR$. I know it appears to be the same because of circular arrangement. Though, I want to overcount the symmetries as much as possible so that we can divide in one go. This leads us to $5!times 5!times 2$. Now each arrangement is over counted $10$ times ($10$ rotations). This leads to my answer $dfrac{5!times 5!times 2}{10}$. I believe this wrong because for simple case of $2$ Democrats and $2$ Republicans, this approach leads to $2$ arrangements. But, it can be easily verified there are $4$ possible arrangements. Can someone please help me find out the mistake and correct answer?