Unlike folding, protein aggregation is a multipathway, kinetically controlled process yielding different conformations of fibrils. The dynamics and determinism/indeterminism boundaries of misfolded conformations remain obscure. Here we show that, upon vortexing, insulin forms two distinct types of fibrils with opposite local chiral preferences, which manifest in the opposite twists of bound dye, thioflavin T. Occurrence of either type of fibrils in a test tube is only stochastically determined. By acting through an autocatalytic, "chiral amplification"-like mechanism, a random conformational fluctuation triggers conversion of the macroscopic amount of insulin into aggregates with uniformly biased chiral moieties, which bind and twist likewise the achiral dye. Although a convection-driven chiral amplification in achiral systems, which results in randomly distributed excesses of optically active forms, is known, observation of such a phenomenon in misfolded protein built of L-amino acids is unprecedented. The two optical variants of insulin fibrils show distinct morphologies and can propagate their chiral biases upon seeding to nonagitated insulin solutions. Our findings point to a new aspect of topological complexity of protein fibrils: a chiral feature of hierarchically assembled polypeptides, which is partly emancipated from the innate left-handedness of amino acids. Because altering chirality of a molecule changes dramatically its biological activity, the finding may have important ramifications in the context of the structural basis of "amyloid strains".