A curious clinical trial design - What is the type / name of this rare methodology? There is this split-mouth randomized clinical trial, in which one side of the mouth receives a particular treatment, and the other side receives another treatment (usually used as the control treatment); (we do this after randomization of the left/right sides of the mouth to treatments A and X [most likely X is a control treatment]).
This is a very good type of randomized clinical trial, because the two sides of the mouth are almost perfectly matched with each other in terms of almost every variable.
Now imagine that we combine three of these split-mouth clinical trials into one single study: We have three groups of subjects (randomly assigned to each group): In each person, one side of the mouth is assigned to the shared control treatment, and the other side is assigned to the experimental treatment. But we have 3 different experimental treatments in the three groups.
All of the three intervention groups have the same control treatment; but each of the 3 groups has a different experimental treatment.
Like this:
Group 1: Side 1: Experimental treatment A -------- Side 2: Control treatment X
Group 2: Side 1: Experimental treatment B -------- Side 2: Control treatment X
Group 3: Side 1: Experimental treatment C -------- Side 2: Control treatment X

What will be the name of this strange and rare type of randomized clinical trial?
And by the way, subjects are randomly assigned to the groups 1, 2, and 3.
 A: The split-mouth design is an example of a within-subject study design. Your extension is a study design with an additional between-subject factor. You don't need to give it a name. From statistical perspective, a "split-mouth" design is exactly the same as a study where two different conditions are tested on the same subject.
The primary endpoint $Y$ of such design is the within-subject difference between experimental side and control side. This is sweet as it removes one of the two experimental factors.
Since your research hypothesis has multiple aspects, it is recommended to care about multiple testing. One possibility is to apply a hierarchical testing approach (see e.g. pdf in https://www2.cscc.unc.edu/impact7/system/files/Bretz_Slides.pdf, Bretz is the reference for hierarchical testing).
More specifically:

*

*Use e.g. Wilcoxon's signed rank test on the pooled data to see if the primary endpoint $Y$ tends to be positive/negative at level $\alpha$.

*If 1. is significant, use e.g. a Kruskal-Wallis test on $Y$ to see if the three experimental treatments are systematically different. This is also done on the level $\alpha$

*If 2. is significant as well, use e.g. pairwise Wilcoxon rank-sum tests to see which experimental treatments are systematically different from each other, using a Bonferroni-Holm adjustment to stick to a family-wise error rate of $\alpha$.

If done in this sequence, your overall composite research question mentioned in the comment will stick to an overall error rate of $\alpha$. Important is that the steps are done conditional on the previous step to be significant. So if e.g. the first test fails, you could not check the second step without inflating the error probability of the first kind.
A problem that I often see in practice is how to handle partial dropouts? I.e. patients with only one side measured.
Edit: naming
If I would need to give the design a specific name, I would probably go for something like "randomized three group split-mouth study" and then explain what it means.
A: Randomized Blocks Design, with two experimental (or treatment) units in each block. People are the blocks. Sides are the treatment units.
A very similar design is used in experiments with sheep, since sheep are one of animals that have twins. Twins have similar or identical genetic makeup, so it is natural to take advantage of this property in doing experiments on sheep.
Mouths differ from sheep in having a left side and right side, but sheep just come in pairs. The left and right sides should be balanced over treatments, that is, half the treatments should be on each side. Exact balance makes the analysis easier.
It is unclear whether the treatments, A, B, and C, are meant to be compared, but it seems desirable or these experiments would just be analyzed separately with paired tests of treatment vs. control.
If the treatments are meant to be compared then it would be efficient (use fewer subjects and get the answer faster with less expense) to think of this as four treatments, A, B, C, and X (the control). Divide the subjects (mouths or blocks) into six groups according to every possible pair of the four treatments: treatments A and B, treatments A and C, treatments B and C, treatments A and X, treatments B and X, and treatments C and X.
All the comparisons can be made among the four treatments, A, B, C, and X (control) with an omnibus ANOVA and then with appropriate comparisons among A, B, C, and X. A book on Design of Experiments can be consulted for the appropriate analysis.
The analysis of the design as described might be given in the literature on experimental design or in agriculture.
There is another factor, Side (Left and Right) that presents an interesting twist and could be tested if desired. One expects the test for this to be insignificant, but some people are sided for chewing and dentists can usually (or often) identify this on examination. If each group of subjects, A, B, and C, is balanced, with an equal number being treated on left and right then an aggregate test of left versus right should be simple and informative.
