A. - wow. Just - wow.
I didn't get the feeling from your post that you're actually friends with this mom, which may make it a little easier to talk to her. You DO need to talk to her, but yes, I agree, it's touchy. It depends on your personality, really, on what's the best way to handle.
If it were me, after I freaked out and then got over my anger, and realized I had to be nice - this isn't TV where you can reconcile easily after confrontations...I would package up my kids, walk next door unannounced, go in for a friendly visit - and tell her "I really like this arrangement - being able to drop our kids off with each other, like how you did the other day. It makes me feel good to know I've got some help nearby when I need a break." (no, it doesn't matter if you'd ever drop your kids off...not the point) "But when you drop Suzy off, can you make sure to touch base with me? If you're going to be actually leaving your house, I'll need to get your carseat from you in case I need to go somewhere. And oh - I don't know if you have my cell number - and I need yours - that way we can keep in contact if we need to."
Maybe something like that would do it. By saying it like that, you're taking away any hostility and can play it from the angle that this is what you thought the deal was - she'll let you know if you're wrong! Who knows what she was thinking when she did that. For all you know, her daughter told her that you invited her to stay all day. My son tells me fantastic lies like that all the time!
So give the mom the benefit of the doubt, have a conversation with her similar to what I laid out above, and then just see where it goes. You'll know REAL quick her true character once you hear how she responds to you, and then you make other decisions from there.
She could be a total slacker mom, which in that case you need to create very strict boundaries and not be so concerned about being friends, or she could be a really lonely, shy mom who just doesn't know how to make connections - either way, you owe it to yourself and the kids' friendship to approach it open-minded and go from there. Good luck!