We calculate the vacuum to meson matrix elements of the dimension-4 operator (psi) over bar gamma(4)(D) over left right arrow (i)psi and dimension-5 operator (psi) over bar epsilon(ijk)gamma(j)psi B-k of the 1(-+) meson on the lattice and compare them to the corresponding matrix elements of the ordinary mesons to discern if it is a hybrid. For the charmoniums and strange quarkoniums, we find that the matrix elements of 1(-+) are comparable in size as compared to other known q (q) over bar mesons. They are particularly similar to those of the 2(++) meson, since their dimension-4 operators are in the same Lorentz multiplet. Based on these observations, we find no evidence to support the notion that the lowest 1(-+) mesons in the c (c) over bar and s (s) over bar regions are hybrids. As far as the exotic quantum number is concerned, the nonrelativistic reduction reveals that the leading terms in the dimension-4 and dimension-5 operators of 1(-+) are identical up to a proportional constant and it involves a center- of- mass momentum operator of the quark-antiquark pair. This explains why 1(-+) is an exotic quantum number in the constituent quark model where the center of mass of the q (q) over bar is not a dynamical degree of freedom. Since QCD has gluon fields in the context of the flux tube which is appropriate for heavy quarkoniums to allow the valence q (q) over bar to recoil against them, it can accommodate such states as 1(-+). By the same token, hadronic models with additional constituents besides the quarks can also accommodate the q (q) over barq center-of-mass motion. To account for the quantum numbers of these q (q) over bar mesons in QCD and hadron models in the nonrelativistic case, the parity and total angular momentum should be modified to P = (-)(L+1+1) and (J) over right arrow = (L) over right arrow + (l) over right arrow + (S) over right arrow, where L is the orbital angular momentum of the q (q) over bar pair in the meson.