Locally irregular graphs are graphs in which no two adjacent vertices have the same degree, while locally irregular decompositions are edge-partitions of graphs into locally irregular graphs. These notions were introduced as a possible antonym to graph regularity, and appeared naturally along the study of related objects and problems, such as proper labellings, other types of distinguishing labellings, and the 1-2-3 Conjecture. In the current work, we introduce and study a stronger version of locally irregular graphs and decompositions, which we call strongly locally irregular graphs and decompositions, where the notion of irregularity is tweaked, in that we require adjacent vertices to have degrees differing by at least~$2$. We prove results of multiple natures, showing both similarities and discrepancies between our modified notions and the original ones. In particular, we prove that graphs admitting strongly locally irregular decompositions are not easy to characterise (unless \textsf{P}=\textsf{NP}), which is not the case for locally irregular ones. Among other results, we also establish bounds on the least number of strongly locally irregular graphs decomposing certain graph classes, and compare them with similar bounds known for locally irregular decompositions. While some of our bounds do not differ much with some for the original context, some others do significantly. We also establish, for our notions, \textsf{NP}-hardness results that go beyond such results known for the original notions.