[en] We present a catalog of early-time ($\sim <!--\; ???\; -->{10}^2-<!--\; ???\; -->{10}^4$ s) photometry and polarimetry of all gamma-ray burst (GRB) optical afterglows observed with the RINGO2 imaging polarimeter on the Liverpool Telescope. Of the 19 optical afterglows observed, the following nine were bright enough to perform photometry and attempt polarimetry: GRB 100805A, GRB 101112A, GRB 110205A, GRB 110726A, GRB 120119A, GRB 120308A, GRB 120311A, GRB 120326A, and GRB 120327A. We present multiwavelength light curves for these 9 GRBs, together with estimates of their optical polarization degrees and/or limits. We carry out a thorough investigation of detection probabilities, instrumental properties, and systematics. Using two independent methods, we confirm previous reports of significant polarization in GRB 110205A and 120308A, and report the new detection of $P=6_{-<!--\; ???\; -->2}^{+3}$% in GRB101112A. We discuss the results for the sample in the context of the reverse- and forward-shock afterglow scenario, and show that GRBs with detectable optical polarization at early time have clearly identifiable signatures of reverse-shock emission in their optical light curves. This supports the idea that GRB ejecta contain large-scale magnetic fields, and it highlights the importance of rapid-response polarimetry.