For the sake of the kids, I would be inclined to give them enough for a token gift - That way you take away his ammunition for badmouthing you to the kids later, and the kids see you as the super-mum that you really are.
My STBX has given our kids money to buy her presents. I will do the same for them to buy me presents.
This is the 3rd xmas since i split with him and he has never once given the children anything to give to me , so i return the favour and give the kids nothing to give to him.Had the split not been like this and my ex was some what normal then i would have suggested both of us giving something to the children to give to us.Having said that i know for a fact my daughter has made me something dunno what it is but that for me is so much nicer than being bought a gift i ain't gonna use and with that added personnel touch.
I think it depends on the situation. The bottom line is that you don't want the kids to be in a situation in which they have nothing to give to a parent to whom they want to give something. So probably the best thing to do is to discuss it with them, unless they are very young. Some new partners will be pro-active in helping the kids with this, and if they do I would welcome it, but some won't get it as an issue, won't realise, or won't bother. Each situation is different.
If necessary, as Mike says, you give the kids money to get you a present - then they still get to do the choosing and they have something to give you. Enlist a friend to help, if necessary. As they get older you may need to give them a general sub to help them through Christmas in any case - once they are in their teens this is probably better than taking them out individually to buy presents for everyone else.
When it was my birthday I gave someone else some money to take the kids out to get me a present. For Christmas my mum is organising it. Grandparents can be very helpful in that respect, especially with little ones.
So not only has she tried to ruin my life, lost me my dream home, daily contact with my children, left me in debt and spread lies against me and my family I am supposed to buy her a christmas present???