Rotating flow inside an internal combustion engine cylinder is deliberately engineered for improved fuel-air mixing and combustion. The details of the rotating flow structure vary temporally over an engine cycle as well as cyclically at the same engine phase. Algorithms in the literature to identify these structural details of the rotating flow invariably focus on locating its center and, on occasion, measuring its rotational "strength" and spatial extent. In this paper, these vortex parameters are evaluated by means of complex moments, which have been adapted from image (scalar field) recognition applications to two-dimensional flow pattern (vector field) analysis. Several additional detailed characteristics of the rotating flow --- including the type and extent of its deviation from the ideal circular vortex, its rotational and reflectional symmetry (if exist), and thus its orientation --- are also shown to be related to the first few low-order complex moments of the flow pattern. The introduction of complex moments as an organizing framework for vortex identification and characterization, therefore, constitutes the major contribution of this paper. The analysis tool is applied to a set of in-cylinder flow fields obtained by high-speed particle image velocimetry at mid-intake stroke in the middle tumble plane of a research optical engine. The cycle-to-cycle variations of the large-scale tumble vortex characteristics --- in location, strength, and orientation --- are quantified and discussed.