We consider the problem of sorting a circular permutation by super short reversals (i.e., reversals of length at most 2), a problem that finds application in comparative genomics. Polynomial-time solutions to the unsigned version of this problem are known, but the signed version remained open. In this paper, we present the first polynomial-time solution to the signed version of this problem. Moreover, we perform experiments for inferring phylogenies of two different groups of bacterial species and compare our results with the phylogenies presented in previous works. Finally, to facilitate phylogenetic studies based on the methods studied in this paper, we present a web tool for rearrangement-based phylogenetic inference using short operations, such as super short reversals.