  Inversion sequences are integer sequences $e=e_{1}e_{2}\dots e_{n}$ such that
$0\leq e_{i}<i$ for each $i$. The study of patterns in inversion sequences was
initiated by Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck in the
classical (non-consecutive) case, and later by Auli--Elizalde in the
consecutive case, where the entries of a pattern are required to occur in
adjacent positions. In this paper we continue this investigation by considering
{\em consecutive patterns of relations}, in analogy to the work of
Martinez--Savage in the classical case. Specifically, given two binary
relations $R_{1},R_2\in\{\leq,\geq,<,>,=,\neq\}$, we study inversion sequences
$e$ with no subindex $i$ such that $e_{i}R_{1}e_{i+1}R_{2}e_{i+2}$.
  By enumerating such inversion sequences according to their length, we obtain
well-known quantities such as Catalan numbers, Fibonacci numbers and central
polynomial numbers, relating inversion sequences to other combinatorial
structures. We also classify consecutive patterns of relations into Wilf
equivalence classes, according to the number of inversion sequences avoiding
them, and into more restrictive classes that consider the positions of the
occurrences of the patterns.
  As a byproduct of our techniques, we obtain a simple bijective proof of a
result of Baxter--Shattuck and Kasraoui about Wilf-equivalence of vincular
patterns, and we prove a conjecture of Martinez and Savage, as well as related
enumeration formulas for inversion sequences satisfying certain unimodality
conditions.
