I don't know if those large and vigorous vines would do, but all of them perform perfectly in part shade:
-Hydrangea anomala ssp. petiolaris,
-Schizophragma hydrangeoides (a lot of new different and exciting new cultivars on a market),
-Akebia quinata,
-Actinidia kolomikta (hardy kiwi).

Zephirine Drouhin cl. rose will bloom in part shade, especially if it's bright in May-early June but don't expect much of rebloom in such position.

I think that both Hydrangea anomala and Schizophragma hydrangeoides are clingers, not twining vines, and aren't really great on trellisses - better on stone walls or tree trunks.

Is clematis less appealing in winter than a climbing rose is?

You are right, I missed that point about trellises.
Anyway I think they will be eating that garage alive.

I don't know much about climbing roses then. I thought they left a tangle of brambles in the winter. I will try to be more open minded about clematis. Maybe I'll go visit the clematis expert in our town.

http://www.clematisnursery.com/

Once I figure out how to post a picture, I'll post a shot of what the garage and trellis look like.

You're right, rose canes stay there all winter, but so do clematis stems. I've got climbing roses on a fence, in nearly full sun. In winter, the bare branches have a certain gnarly charm, I suppose, and the rose hips are appealing while they last.

You might consider jasminum nudiflorum, which has very early blossoms and stems that stay green all year. It needs to be tied or woven through its support, like roses, as it's not a twiner. Mine's in fairly constant shade, but blooms very well. The best thing about this plant is that it blooms in winter!

There's also carolina jessamine (Gelsemium sempervirens), a semi-evergreen vine that blooms in early spring. It's only hardy to somewhere between 0 and 10 degrees, so it might need protection where you live. My jessamine gets hot afternoon sun, but I've read that it takes quite a bit of shade.

The seed-heads of sweet autumn clematis stay in place for quite a long time, and the new growth stays green through most of the winter, but I can't recommend this vine because I've been working so hard to eradicate it from my yard - it's highly invasive, even in shade. There are other vigorous clematis varieties that are much better behaved, like C. montana.

Some varieties of honeysuckle are nearly evergreen, but I don't know of one (there might be one) that's not too invasive but looks good most of the year.

Or, go with roses, if that's what you'd really like to see on the trellis. It might be worth a try!

The Hydrangeas attain massive size (60-70+' over time) and they do cling. I have one.
Roses will likely not flower well for you in a shaded location. Clematis might, but select your variety carefully. Many can be cut about 12" from the ground in early spring and fill and entire trellis effortlessly in a season and flower, too.

You might consider Aristolochia durior (Dutchman's pipe), though it isn't generally noted for flowers, more for the huge, heart shaped leaves. You can cut this back hard, too and it will get really big. This is the variety you so often see on Victorian trellises.

Give the smarties over to the Vines forum a holler... they'll have lots of good things to suggest.

I really appreciate all your ideas. I do realize that Clematis is probably the answer. I do have a Clematis specialist in our town. Hydrangea is another option although a little slower growing.

I'll post a picture of the side of the garage with the trellis later tonight. You'll maybe agree with me that a trellis was called for in combination with the rock-sided garage. So yes, I did put in the trellis and then went looking for the plant. I guess I was confident there would be an answer. It wasn't the one I wanted - roses - but I have another spot for that. I guess that'll be next years project. :-)

I have Carolina jesamine that I bought at Garden in the Woods last fall. It has been slow starting, but that is likely because of the "sleep, creep, leap" process. It overwintered with no problems, perhaps because of the snow insulation. But, the info said that it has been proven hardy in zone 6, and Garden in the Woods even suggests that it could be hardy in zone 5 with protection.

How fast of a grower are you looking for?
Polygonum aubertii
Silver Lace Vine
Zone 5
Extremely vigorous vine. May grow 25' in a single season. Ornamental foliage, covered with large clusters of fragrant, white blooms in late summer and fall. Creates a wonderful frothy effect. For sun to part shade, in well-drained soil. Prune drastically in late fall. Mulch with evergreen boughs in exposed locations. Can grow 25-35' or more!

Here is a link that might be useful: silver lace vine (not the best picture of one)