  The study of information revivals, witnessing the violation of certain
data-processing inequalities, has provided an important paradigm in the study
of non-Markovian processes. Although often used interchangeably, we argue here
that the notions of ``revivals'' and ``backflows'', i.e., flows of information
from the environment back into the system, are distinct: an information revival
can occur without any backflow ever taking place. In this paper, we examine in
detail the phenomenon of non-causal revivals and relate them to the theory of
short Markov chains and squashed non-Markovianity. As a byproduct, we
demonstrate that focusing on processes with actual backflows, while excluding
those with only non-causal revivals, resolves the issue of non-convexity of
Markovianity, thus enabling the construction of a convex resource theory of
genuine non-Markovianity.
