It's tricky. I think if the behavior was severe (getting suspended from school, taking the car without permission, etc.) then the consequence can be enforced on both sides **with prior discussion about what that consequence might be**. But if she lied to her mom, then she needs to face the music with her. You can back her up by saying that you are disappointed with this behavior and that she should not lie to her mom. But this kind of falls in the gray area of how much control do you have in each other's homes? If there's no dietary concerns, but you don't like her having sugar, does that mean she can no longer buy Froot Loops? Or if it works for their household to have her bedtime at 8:30 but she sees you mostly on weekends do you have to enforce that bedtime?
What one of my friends did was make a clock. The kid would act up right at the end of visits, figuring she'd be scott free in a couple of hours. So SM and Dad said no, next time we see you, you owe us. You owe us the 10 minutes you fought about putting on your shoes and getting out to the car. You owe us the 10 minutes you argued in the store. Etc. And the next time she came over, they worked out how she paid back the time on her clock. Maybe she sat out for 20 minutes while her cousins played. Or maybe she did 20 minutes worth of chores or went to bed early. Friend said it really stopped a lot of end of the weekend fights and they got to enjoy their time much more.
DH can also say, "I back you up that she needs to stop x behavior. However, we have family time planned and I will have to think about how I'm going to deal with her behavior on my time." So maybe your household goes to the movie, and she has to do something to fix the problem with her mom. Write a letter of apology or something.
It's just a slippery slope when you dictate each other's households, especially if there are very different parenting styles. It's kind of like "just wait til your father comes home!" which means Mom just gave up her authority and Dad is always the heavy. And that's no fun, either. It is not uncommon for a kid to act one way in one place and another way in another place. They will behave according to what they can get away with/what the expectations are. Mom's house, Dad's house, Grandma's house, school...
IMO, there's more than one way to be united on the end goal of not tolerating the behavior, even if you take different steps to get there. And, also, if there's a behavior seen often in one home and not in the other, then the change needs to happen in A and not necessarily B. You can try to help her through it on your end, but Mom needs to change something on hers if SD predominantly lies to HER. If my SD didn't pick up her bathroom at her mom's house, there was nothing we could do about it. That was on her mom and SF.
Bottom line is IF they are going to do this, they need to discuss each situation in advance and agree to terms - which may or may not be exactly the same in each home, depending on parenting styles. Mom may be angry, but that doesn't mean SD comes to your home every time with a list of consequences you were not involved in making.