A single 2-way is hard enough. A double 2-way is confusion waiting to happen!
The old L1 and L2 on each switch are what we call "travelers". They are a matched pair, and one is hot or the other one is. They are interchangeable. As such, I like to mark them both yellow with tape. I prefer to wrap the tape around the wire as far back as practicable.
Now here, you have two complete sets of travelers, and you must match each pair to their partner common wire. In some cases, this is completely obvious, e.g. Both travelers and common are in the same cable. Other times, it doesn't matter because both commons are split from the same wire. **But if it's any other thing, mark the commons also in some way to associate each with its travelers.
For instance you might mark one set of travelers blue and the other yellow, and then mark their common with a half-width piece of tape of the same color.
While the old switch is still on, marking will seem like a useless chore. Why not just move the wires over, one wire at a time, and not make any mistakes, and not get confused at all? Heh... You know what happens there. It's practically inevitable. And honestly, most likely we would not be able to help you much at that point. It would take some fairly complex probing.
Anyway, on the old switch, you can tell how it's split - C is common, L1 and L2 are the travelers.
On the new switch, Lx is common, and Lx1 and Lx2 are the travelers. X is 1 or 2, of course.