Hi, good question but the important thing to know is that basically the MPS and MPO classes in ITensor are essentially just arrays of ITensors. The only details beyond them just being arrays are that they "remember" their orthogonality center (if they have one) and they have some associated functions that expect those types (MPS, MPO) to be passed.
But you can always put any collection of ITensors into an array and pass this array around, or store this array in a custom type (MPDO, say) then define analogous functions or operations on it as we have for MPS and MPO.
Finally, an MPDO is structurally equivalent to an MPO, correct? If so you could just repurpose the ITensor MPO type for your work. Or are you referring to the network I know as a "locally purified state" which looks more like two MPO's contracted together?