There's two ways to say: "The invigorating wind lashed her face"

1) Le vent revigorant lui fouettait le visage.
2) Le vent revigorant fouettait son visage.

The second one is how you would also say it in English. It is grammatically correct, but the first one is the more "French" way to say it. Literally, it is: The invigorating wind lashed her the face. Here, there is no possessive pronoun (son) but possession is expressed via the indirect object pronoun lui.

Ce qui donnait à ses joues une douce teinte rosée. -- Which gave his cheeks a soft, pink tinge.
Ce qui lui donnait une douce teinte rosée. -- Which gave him a soft, pink tinge.

I am not aware of any rule that requires the use of reflexive with body parts.

The usage is donner qqc à qqn. -- to give something to someone.

hi  Chris; 

A different but similar sentence  - The invigorating wind lashed her face, is translated by the Quizic team - Le vent revigorant lui fouettait le visage:

Looking again I can see this is not reflexive, however in this example the action is not being done directly to the face but to her (expressed as lui) -  Lui fouettait les le visage -  and the face (le) is acted on indirectly.

Do you know  why there is a difference in the way this sentence are written?

I think I commented on that in my previous post. It is the same pattern:

Ce qui lui donnait une douce teinte rosée. -- Which gave him a soft, pink tinge.
Le vent revigorant lui fouettait le visage. -- The invigorating wind lashed his face. ("lashed him the face")

Hi Chris 

Yes, indeed 'Ce qui lui donnait une douce teinte rosée'  translates 'Which gave him a soft, pink tinge'.

But please look again I what I actually wrote i.e 'ce qui lui donnait les joues une douce teinte rosée'?