The critical behavior of the three-dimensional N-vector chiral model is studied for arbitrary N. The known six-loop renormalization-group (RG) expansions are resummed using the Borel transformation combined with the conformal mapping and Padé approximant techniques. Analyzing the fixed-point location and the structure of RG flows, it is found that two marginal values of N exist which separate domains of continuous chiral phase transitions N>N_c1 and N<N_c2 from the region N _c1>N>Nc2 where such transitions are first order. Our calculations yield N_c1 = 6.4(4) and N_c2 = 5.7(3). For N>N_c1 the structure of RG flows is identical to that given by the ε and 1/N expansions with the chiral fixed point being a stable node. For N<N_c2 the chiral fixed point turns out to be a focus having no generic relation to the stable fixed point seen at small ε and large N. In this domain, containing the physical values N = 2 and N = 3, phase trajectories approach the fixed point in a spiral-like manner giving rise to unusual crossover regimes which may imitate varying (scattered) critical exponents seen in numerous physical and computer experiments.