The dance style I'm referring to is the one where each partner presses one palm to the others' and they move in circles while maintaining eye contact (and exchanging flirtatious banter). Is this just a trope or does it have a historical basis?
Factual basis yes. Absolute proof no. We start to have a very good idea of how people danced and dance's from about 1500 on. The documentation from the medieval period is mostly based around court dances, so it's hard to say how the common people danced, but we can trace the steps of the nobility much better then one would expect. It should also be said that "Palm to Palm" is not a feature of every courtly dance, but a specific move in dances that are more about making complicated shapes then dancing in two rows. I've been taking a historical dancing class for the last year and a half in England for my MA.
It's not just modern movies and TV—at first I thought you quoted "Romeo and Juliet":
JULIET: Good pilgrim, you do wrong your hand too much, Which mannerly devotion shows in this; For saints have hands that pilgrims' hands do touch, And palm to palm is holy palmers' kiss.
ROMEO: Have not saints lips, and holy palmers too?
JULIET: Ay, pilgrim, lips that they must use in prayer.
ROMEO: O, then, dear saint, let lips do what hands do; They pray, grant thou, lest faith turn to despair.
JULIET: Saints do not move, though grant for prayers' sake.
ROMEO: Then move not, while my prayer's effect I take. Thus from my lips, by yours, my sin is purged.
JULIET: Then have my lips the sin that they have took.
In this scene, they explicitly dance 'palm to palm'. Does this mean that it was there during Medieval times? It's hard to say, especially since BECAUSE of romeo and juliet the act has been copied and alluded to many times, and on top of that we have a rich visual culture of recordings now to build on for these dances, so it's easy to retroactively impose this visually stimulating situation onto a dance where lovers touch palms and talk.