You need to consider your climate in careful detail, since few evergreen climbers are fully hardy in every part of the country, except ivy. Probably the evergreen honeysuckles, such as Lonicera henryi, are the hardiest choice. Some others are commonly grown, such as trachelospermum spp (my neighbour's was undamaged by -10C, but that must be pushing it close), and clematis armandii,(my brother's was killed by last winter, that in mildish Warrington). Akebia quinata is pretty hardy, but only evergreen in a mild winter. Berberidopsis corallina, Lapageria rosea and Abutilon megapotamicum all died at -10 in my garden last winter, having survived -9 the previous year. Passion flowers are an option, but again not strictly hardy. Holboellia latifolia is probably similar. Chilean potato vine is probably rather less hardy. Although not strictly a climber, there are some varieties of Pyracantha which are grown as wall shrubs and are evergreen and hardy. Maybe there are some other not-really-climber evergreens which can be grown as wall-shrubs in a similar way.

There you are, some ideas, but you will need to research what is likely to survive your winters.