Overwintering perennials in pots has come up before here and it's evoked some disagreement (e.g. below). What's worked for me, at least here (Z5b), has been leaving a wide variety of herbaceous perennials, potted in garden soil, outside over winter. Over winter the soil in the pots remains frozen solid. There's been some losses, but most of (several hundred) plants I've overwintered in this way have survived.

https://www.houzz.com/discussions/5386615/perennials-in-pots-winter-care-help

It's a non-issue in my climate :-) But you should be aware that the roots are the most vulnerable plant parts to cold temperatures and containerized plants are exposed to far colder temps than they would be if in the ground. Soil is a great insulator :-)

If the plants were valuable to me, I doubt I'd risk leaving them outside unprotected all winter, even if tilted on their side. I'd either group them altogether and bury thickly in straw or dried leaves or haul them into a garage or other cold but not freezing area for the winter.

Certainly relevant points. I have always left the overwintering potted perennials in a protected location, but I'm thinking that a key part of that protection is keeping them out of the sun (viz. during a mid-winter (January) melt). Once frozen, they should stay frozen until the spring thaw when it's time to move them out into the spring sun.

OK I really don't want to risk losing them, so I will keep them in my unheated attached garage or the shed. What do I do about watering though? When I kept my other annuals in my room temperature basement last year, they were bone dry all the time so I'm sure that's why I lost them. Where do I draw the line between too much water/not enough?

re: mandevilla, I would be OK with leaving it inside but my window light may not be enough? I have grow lights in the basement I can keep those on?

A room temp basement is too warm :-) As ken noted, these need to go dormant and stay dormant and that typically involves temps in the 35-45F range. And you want the container soil to stay just barely moist, not very wet and never completely dry.

There are various online articles describing how to overwinter mandevilla. There are two primary methods: one that involves light and a second that does not :-)

Re: watering. Make sure the pots are well-watered before pulling into storage. Once they freeze (and they will if in unheated garage or shed), you don't water until they thaw. One trick I use is I pack snow on them mid-winter, then when the temps warm enough to melt the snow it slowly, gently waters the pots. Works great. Pull out of storage when temps warm up a bit -- they will start growing on their own timetable once they thaw, so spring is a constant chore of pulling outside during the day for the light and back outside for the night if frost is on the way. This is for hardy perennials (not the mandevilla)

Great! So, would you say I should start bringing them in the shed when I'm seeing night time temps at 35-45? Then start transitioning them out when I'm seeing temps in the..... what range?

No - bring them in when they are fully dormant and temps in the 30s-high 20s range. They're going to freeze solid regardless -- temps in the 20s won't kill them -- and you want them to be completely dormant before pulling them into storage.