I am not sure about the evergreen part but I did notice in my garden it did remain to stay green all year. Wild Blue Phlox, Phlox divaricata it blooms in the spring a pretty blue, very fragrant. Phlox also reseeds it self so it should spread quite nicely. Here in my area its a wonderful site to see the phlox growing in the wild on slopes and hills amongst the rocks and trees. Its height is about a foot tall so its not your typical ground cover but I know from what I have seen its a beautiful thing for hillsides when its in bloom. I added a link so maybe this will give you an idea of the look.

Here is a link that might be useful: USDA plant profile

A low growing juniper may also work; your local nursery can help you with varieties that do well locally. Some juniper species are shade tolerant.

Vinca is a bad word. It may look nice but takes over an area fairly quickly. Under my spruce tree must be two hundred or so vinca seedlings that are just coming up. Think how many seedlings could come from as large of an area as needed covered.

Euonymous is another that can spread.

If you really want invasive, check out Yellow Archangel.

Have a containment plan, though.

Depending on the height here a few options to research. Lecothoe, Archtostaphylos, Gaultheria, Sarcococca, Microbiota decussata, Skimmia, Mahonia repens, Euonymus 'Harlequin', Vinca, Pachysandra. It might be a good idea to use a combiantion to add a little dimension to the spot.Also don't be afraid to add a little vertical interest, Pieris, Mahonia japonica or Mahonia aqufolium, Kalmia, are just a few examples.

You may be using the word "invasive" to me something other than it's actual scientific meaning. But, if you really are looking for an invasive plant, you are doing your yard, your community, and the environment an extreme disservice! Planting invasive plants is a very damaging form of littering. Invasives are not only litter, but they are a form of litter that increases by itself over time. They are harmful to native plant and animal life and cost this nation hundreds of billions of dollars per year!

Call an online bamboo nursery to ask about a small low growing variety for your slope.
There is nothing better than a grass, which bamboo is, that will be evergreen and will also help keep the soil on your slope which is another consideration.
You want a low growing, pretty bamboo for ground cover.
Tell them that, they will recommend something for you.
All bamboo is evergreen.
Good Luck to you!

Spreading bamboos may work well if you are able to keep their growth in check. Most are either not invasive or only marginally invasive in most climates. You would not however want to plant an aggressive bamboo next to a property line where it could become a nuisance to one of your neighbors.

Not all bamboo is evergreen. Borinda emeryi and Borinda frigidorum are two examples of normally deciduous bamboos. Also, many bamboos are facultatively deciduous in many climates or conditions.