You can also report that direct statement by Joe afterwards as follows:
Joe said he would clean his room when he got home.
Whether there will be a shift of tense depends on the context.
Let's say two parents are bickering over their son's messy room. Joe is on the phone (speaker phone), and the parents are speaking with him in a three-way conversation at this moment:
Dad: Joe, your room is a terrible mess. There are
potato chips all over the floor, and one of your textbooks has Coca Cola
spilled all over it.
Joe: Dad, I will clean my room when I get home. I am going to a movie
with my friends first.
Dad: I want you to come home from wherever you are and clean this room
Mom: Let him go to the movies with his friends. Joe said he will clean his room when he gets home.
Now let's say the parents are speaking about that incident a week later.
Mom: When I said that we should let Joe go to the movies with his
friends, you got very angry.
Dad: He said he would clean his room when he got home. I wanted him
home that very minute! We work very hard to
afford this place, and he seems to have no respect or appreciation for
If the reported speech is simply echoing what has just been said a moment ago, as in the first scenario, there might be no shift of tense, or there might be a shift of tense. It is when the speaker thinks of the utterance as a past utterance that the shift happens. The perception of pastness could happen instantly:
He just said a second ago that he would clean his room when he got
In the first scenario, Mom is Joe's advocate, and that role could also affect her choice of tense subconsciously. She is, in effect, repeating Joe's words quasi-verbatim as Joe's proxy in the argument with Dad.