One dimensional binary string with periodic boundaries and reflection

Question

I have a binary string $l=(l_1,l_2,\ldots,l_{2n})$ with $l\in\{0,1\}$ and the conditions $l_i \cdot l_{i+n}=0$ for all $i$ and $\sum l_i=n$. Now, I was wondering how many distinct string exist, when a string is equivalent to another string by the transformations $$(l_1,l_2,\ldots,l_{2n})\to(l_{2n},l_1,\ldots,l_{2n-1}),\\(l_1,l_2,\ldots,l_{2n})\to-(l_1-1,l_2-1,\ldots,l_{2n}-1)?$$ Are there symmetry groups which are covering and already solved this problem? I'd be very grateful for a hint.