Those things are monsters. You could cut it to a stump and it would resprout. How do I know? That's what mine did.
:-)

Well, cutting back to a stump may be a bit severe. Typically if you cut back a woody evergreen vine to only its woody base, you can set it back drastically or even kill it. I had a very large one growing in my old garden that was badly and improperly pruned by the new owners -- last time I drove by it was dead as a doornail. But you can prune back relatively hard with good results but do so as soon as possible after flowering.

Generally, these types of clematis (the early flowering species, aka pruning group I) are listed as "no prune" varieties. That doesn't mean you can't, only that routine pruning (i.e., annual pruning) is not needed or recommended. And they bloom on old growth, so pruning should be done immediately after the bloom cycle.

If yours requires annual pruning to keep in check, it might not be the right plant for that location. Clematis armandii is a BIG vine and should be sited accordingly where it has space to grow unchecked.

Any kind of climber planted to grow in a defined and limited space is liable to need routine pruning and training to produce an appealing effect. The only exception is those growing through trees that are large enough to accommodate all of the growth of the vine. Even then some gardeners might be bothered by the amount of the host tree that disappears beneath the foliage of some vigorous kinds, attempt to do some thinning.

Apart from its vigor and size the main problem with evergreen clematis on structures is the persisting dead leaves that it produces. Annual tidying is required to avoid this.

Of course you can kill a plant by cutting it to a stump. That is exactly what I was trying to do, since mine appeared to have plans to take over the county. The point is that clematis armandii is such a durable plant, cutting it down completely won't necessarily stop it.

These vines are very large and grow rapidly. Feel free to prune a healthy, mature specimen heavily without worries about harming it. They respond well to pruning, which is a good thing because anyone with less than 40 acres or so will likely need to do so at some point. To preserve the most bloom for next year, prune fairly soon after it finishes blooming this year, but in my experience, it doesn't have to be immediately after blooming. I often didn't get around to it until June or so with no ill effects. One year when I pruned in August, I still got a good display the next spring. This isn't what you may read in books about these plants, but it is how mine behaved.

Thanks to all who have responded. I appreciate your information. I will get going on pruning this puppy after flowering stops. My goals are to minimize the dead leaves and wood, and to maximize the living vines, leaves and once a year beautiful fragrant flowers.