This Regiment, which appears to have been unnumbered (although IR122 seems appropriate), was raised in 1758 from the Basel area although most of the officers came from Alsace, perhaps by virtue of their bilingualism.
The other thing that is unusual is the yellow facings. The other 10 Swiss regiments in French service (as well as the Swiss Guards) had blue facings and Kronoskaf suggests that in 1761/2 the facings changed to blue. Meanwhile, for a few short years, the regiment had these striking yellow facings. Figures are from Crusader.