I am interested in knowing what words/phrase/idioms I can use to express a double reversal of decision.
To illustrate that with an hypothetical example to clear any confusion that may have risen :
I had the idea of buying a phone for someone, decided against it, and then thought about it a bit more and ultimately chose to do it.
What do you think about saying "I changed my mind, and then I changed my mind back." in that context?
To come full circle could work. Here are several definitions and examples of the phrase:
To return to the same situation or attitude you originally had. I left publishing, tried teaching, and now I've come full circle back to publishing.[1]
When something “comes full circle,” it completes a cycle, returns to its beginnings: The novelist's vision of human life has come full circle—from optimism to pessimism and back to optimism again. [2]
In the comments below, an idiom describing "returning to a starting point after going around in a full circle[,] especially in the context of unsuccessfully trying to find one's way in a maze" was requested.
I would suggest Going around in circles or Walking in circles. Examples:
To move over and over on a circular path. The model plane went around in circles until it ran out of fuel. The oxen went around in circles, pulling along a beam that was connected to the millstone.[3]
To act in a confused and disoriented manner. I've been going around in circles all day. The children have been going around in circles, waiting for you to arrive.[3]
To keep going over the same ideas or repeating the same actions, often resulting in confusion, without reaching a satisfactory decision or conclusion. We're just going round in circles discussing the problem. We need to consult someone else to get a new point of view. Fred's trying to find out what's happened but he's going round in circles. No one will tell him anything useful.[3]
This expression is frequently used in the context of being lost. Upon realizing that one has returned to a location one has already visited, one might remark, "it looks like I've been walking in circles".
Citations:
1 The Cambridge Dictionary of American Idioms © 2002
2 The American Heritage New Dictionary of Cultural Literacy, Third Edition © 2005
2 McGraw-Hill Dictionary of American Idioms and Phrasal Verbs © 2005
I might suggest that you have flip-flopped on the idea of buying a new phone but ultimately decided to purchase it.
I might also say that you waffled on the idea before buying your new phone.
Consider reconsider.
to consider again, especially with a view to change of decision or action: ex., to reconsider a refusal.
Parliamentary Procedure. to take up for consideration a second time, as a motion or a vote, as with the view of reversing or modifying action taken.
While it does not always mean you will reverse the decision, it means you will think about it again.
I had the idea of buying a phone for someone, decided against it, but on reconsideration chose to do it.
Did a 360.
Two 180s, a Uturn followed by a Uturn.
You can say that you returned to the original idea. This phrase includes the notion of starting with one idea, moving to a different idea, then going back to the first idea.
Here are some examples of this phrase being used (emphasis, mine):
After a failed approach to create a cheaper version of this experiment as part of his diploma thesis in 1930, Ruska returned to the original idea of using short coils as lenses. - nobleprize.org
[Maurepas's] opening correspondence in 1728 rejected the plan of a separate lighthouse and returned to the original idea of a light in the tower over the barracks. - Parks Canada History (kindly suggested by J.R. in comments)
Therefore, we returned to the original idea to implement Asprova in Sulzbach, and unify the system with other manufacturing plants. - AIMagazine
In your case, you returned to the original idea of buying the phone (for "someone").
You can say you have gone back and forth or have oscillated on an issue / on your position.
The verb oscillate can be traced back to the Latin word oscillum, meaning "swing," so it makes sense that oscillate is used to describe an object like a fan or a pendulum that swings from side to side. The word also can be used to describe a different kind of motion — the wavering of someone who is going back and forth between conflicting beliefs or actions. If you’ve ever had trouble making up your mind about something, you probably know what it feels like to oscillate — back and forth from one decision and to another and then back again. And again. And again.
Vocabulary.com (an excellent 'plain language' dictionary)
If you want to emphasize the reversal, you can use Better call the calling off off. That phrase comes from Ira Gershwin's lyrics for Let's Call The Whole Thing Off. The sound of that phrase will be more familiar to the reader, and 'calling off off' is quicker to understand than 'walk back back.'
When you're moving in one direction, you might decide to reverse course and return to your earlier position. This can be called doubling back. This term is often used as a metaphor for returning to a prior opinion after exploring a new alternative opinion and finding it less desirable than the original.
While advancing a line of reasoning, if you pursue a new idea and later decide that idea isn't worth pursuing, you can backpedal and abandon that idea in favor of finding a different one. This term compares the act of ceasing to pursue a new idea with the physical act of pedaling backwards on a bicycle, causing the bicycle to stop (at least for simpler types of bicycle, as were prevalent when the idiom was likely first popularized).
Rather than selecting an expression from a thesaurus for you, I will answer the question
What do you think about saying "I changed my mind, and then I changed my mind back" in that context?
You're almost there.
I changed my mind, and then I went back to my original idea.
By the way, the expression in your title doesn't work in English!
"I did an about face" is also heard in reference to reversing an opinion or course of action.
I do prefer 'to come full circle', as already voted here but if the reason for not going with plan A was perhaps because of a noticeable flaw in it, where plan B at first seemed better but then on reflection perhaps not, you could also consider:
"To 'bite the bullet' is to endure a painful or otherwise unpleasant situation that is seen as unavoidable" - Wikipedia
The key point here is that the 'original idea' is that which was unavoidable.
Perhaps a little stretched with regard to your opinion of a handphone, but ...
You recanted your beliefs/opinions.
to announce in public that your past beliefs or statements were wrong and that you no longer agree with them
‘Galileo was forced to recant his assertion that the earth orbited the sun’
also
say that one no longer holds an opinion or belief, especially one considered heretical. "heretics were burned if they would not recant" synonyms: renounce, forswear, disavow, deny, repudiate, renege on, abjure, relinquish, abandon;
