[RPG] How do the weapon properties Versatile and Finesse work together

Versatile only changes the damage dice you use, and finesse only changes which ability modifier you can use.

Looking at the definitions of finesse and versatile we have:

Finesse: When Making an Attack with a finesse weapon, you use your choice of your Strength or Dexterity modifier for the Attack and Damage Rolls. You must use the same modifier for both rolls.

Versatile: This weapon can be used with one or two hands. A damage value in parentheses appears with the property—the damage when the weapon is used with two hands to make a melee Attack.

Nothing in the rules explicitly says that a weapon can't have both properties, and in that case, the rules don't contradict each other.

In effect, such a weapon would allow you to use it with two hands to change the type of damage dice you roll and also allow you to choose between Dexterity or Strength for your static to-hit and damage modifiers.

There is even an example of this kind of weapon in the 5e SRD: The Sun Blade

This item appears to be a longsword hilt. While grasping the hilt, you can use a bonus action to cause a blade of pure radiance to spring into existence, or make the blade disappear. While the blade exists, this magic longsword has the finesse property (emphasis mine)

Even though the Sun Blade is a rare magic item, it proves that the concept of a versatile, finesse weapon is both possible and intended.