In a multitude of life's processes, cilia and flagella are found indispensable. Recently, the biflagellated chlorophyte alga
Chlamydomonas
has become a model organism for the study of ciliary motility and synchronization. Here, we use high-speed, high-resolution imaging of single pipette-held cells to quantify the rich dynamics exhibited by their flagella. Underlying this variability in behaviour are biological dissimilarities between the two flagella—termed
cis
and
trans
, with respect to a unique eyespot. With emphasis on the wild-type, we derive limit cycles and phase parametrizations for self-sustained flagellar oscillations from digitally tracked flagellar waveforms. Characterizing interflagellar
phase synchrony
via a simple model of coupled oscillators with noise, we find that during the canonical swimming breaststroke the
cis
flagellum is consistently
phase-lagged
relative to, while remaining robustly
phase-locked
with, the
trans
flagellum. Transient loss of synchrony, or
phase slippage
, may be triggered stochastically, in which the
trans
flagellum transitions to a second mode of beating with attenuated beat envelope and increased frequency. Further, exploiting this alga's ability for flagellar regeneration, we mechanically induced removal of one or the other flagellum of the same cell to reveal a striking disparity between the beatings of the
cis
and
trans
flagella, in isolation. These results are evaluated in the context of the dynamic coordination of
Chlamydomonas
flagella.