Choice of iterative solver for a sparse asymmetric matrix with symmetric structure

I have a sparse $nxn$ matrix A with pretty interesting structure. It has a block structure with symmetric structure but asymmetric blocks. Expressed mathematically $A_{jk} = A_{kj}$ but $A_{jk} neq A_{jk}^T$, so my matrix is not actually symmetric. Are there any algorithms that can take advantage of this kind of system?