The azimuthal anisotropic flow of identified and unidentified charged particles has been systematically studied in Cu+Au collisions at root s(NN) = 200 GeV for harmonics n = 1-4 in the pseudorapidity range vertical bar eta vertical bar < 1. The directed flow in Cu+Au collisions is compared with the rapidity-odd and, for the first time, the rapidity-even components of charged particle directed flow in Au+Au collisions at root s(NN) = 200 GeV. The slope of the directed flow pseudorapidity dependence in Cu+Au collisions is found to be similar to that in Au+Au collisions, with the intercept shifted toward positive pseudorapidity values, i.e., the Cu-going direction. The mean transverse momentum projected onto the spectator plane < p(x)> in Cu+Au collision also exhibits approximately linear dependence on pseudorapidity with the intercept at about eta approximate to -0.4 (shifted from zero in the Au-going direction), closer to the rapidity of the Cu+Au system center of mass. The observed dependencies find a natural explanation in a picture of the directed flow originating partly due the "tilted source" and partly due to the asymmetry in the initial density distribution. A charge dependence of < p(x)> was also observed in Cu+Au collisions, consistent with an effect of the initial electric field created by charge difference of the spectator protons in two colliding nuclei. The rapidity-even component of directed flow in Au+Au collisions is close to that in Pb+Pb collisions at root s(NN) = 2.76 TeV, indicating a similar magnitude of dipolelike fluctuations in the initial-state density distribution. Higher harmonic flow in Cu+Au collisions exhibits similar trends to those observed in Au+Au and Pb+Pb collisions and is qualitatively reproduced by a viscous hydrodynamic model and a multiphase transport model. For all harmonics with n >= 2 we observe an approximate scaling of v(n) with the number of constituent quarks; this scaling works as well in Cu+Au collisions as it does in Au+Au collisions.