In music, particularly in the context of chord theory, the terms "root position," "first inversion," "second inversion," and "third inversion" are indeed real names describing the arrangement of the notes in a chord. Each term denotes a specific way of organizing the notes based on which note is the lowest.
If you're looking for a term that is not a recognized name in this context, possibilities include terms like "fourth inversion" or anything outside the traditional naming conventions (like "upside down" or "backwards"). If you have a specific term in mind, I can help clarify whether it is a recognized nomenclature for describing chord inversions.