Nelly Moser and Hagley Hybrid clematis are best planted in shadier conditions, their flower colours fade in bright sunlight. Both are early flowering though, and if they get no sun at all, they may not do too well, or you might find they head for wherever the sun is, and if that's next door, they'll be seeing the flowers over their side of a fence. They like their heads to get some sun. Generally, any flowering climber prefers some sunlight.

If its really shady, there's always Hydrangea anomala or Garrya elliptica 'James Roof'.

Clematis Armandii lovely long pointed leaves, evergreen and when it flowers, it flowers which are beautiful , once established it can quickly cover large areas, good for climbing up trees, but with correct pruning you can keep it in check, with clever planting you can create interest through the year to break up the evergreen effect.