I think one would have to attack it the other way: focus on each vortex first, classify its connectivity. Since I think any path from East to West would break a path from North to South and vice versa, you'd have to classify each vortex by what it connects, down to one of:

    {., NE, NW, ES, SW, NS, EW, NE/SW, NW/SE, NES, NEW, NSW, ESW, NESW}

In a more visual language you'd have 5x5 tiles which the bigger picture is made out of:

    \   /    \            /    \   /    \   /    \  /
     \ /      \          /      \ /      | |      ‾‾
      X        X        X        X       | |
     / \      / \      / \      /        | |      __
    /   \    /   \    /   \    /        /   \    /  \

    \            /             \            /    \   /
     \          /               \          /      \ /
      \        /        X        X        X        X        .
       \      /        / \      /          \
        \    /        /   \    /            \

Once you've figured out each individual vortex I think solving the bigger problem is probably doable?