  This paper is the first from a series of papers that establish a common
analogue of the strong component and basilica decompositions for bidirected
graphs. A bidirected graph is a graph in which a sign $+$ or $-$ is assigned to
each end of each edge, and therefore is a common generalization of digraphs and
signed graphs. Unlike digraphs, the reachabilities between vertices by directed
trails and paths are not equal in general bidirected graphs. In this paper, we
set up an analogue of the strong connectivity theory for bidirected graphs
regarding directed trails, motivated by factor theory. We define the new
concepts of {\em circular connectivity} and {\em circular components} as
generalizations of the strong connectivity and strong components. In our main
theorem, we characterize the inner structure of each circular component; we
define a certain binary relation between vertices in terms of the circular
connectivity and prove that this relation is an equivalence relation. The
nontrivial aspect of this structure arises from directed trails starting and
ending with the same sign, and is therefore characteristic to bidirected graphs
that are not digraphs. This structure can be considered as an analogue of the
general Kotzig-Lov\'asz decomposition, a known canonical decomposition in
$1$-factor theory. From our main theorem, we also obtain a new result in
$b$-factor theory, namely, a $b$-factor analogue of the general Kotzig-Lov\'asz
decomposition.
