The azimuthal anisotropy Fourier coefficients (vn) in 8.16 TeV p+Pb data are extracted via long-range two-particle correlations as a function of the event multiplicity and compared to corresponding results in pp and PbPb collisions. Using a four-particle cumulant technique, vn correlations are measured for the first time in pp and p+Pb collisions. The v2 and v4 coefficients are found to be positively correlated in all collision systems. For high-multiplicity p+Pb collisions, an anticorrelation of v2 and v3 is observed, with a similar correlation strength as in PbPb data at the same multiplicity. The new correlation results strengthen the case for a common origin of the collectivity seen in p+Pb and PbPb collisions in the measured multiplicity range.