  We propose a new way of studying the spin content of a hadron by looking at
its response in a rotating frame. By collecting all responses of quarks and
gluons in a rotating frame, we describe the spin-rotation coupling of spin-1
quarkonia and thereby reveal their spin contents in a relativistic formalism.
We demonstrate that both the perturbative and non-perturbative contributions in
the operator product expansion follow a universal formula that identifies the
spin-rotation coupling with unit strength. This allows us to recognize the
total spin-1 of the vector and axial vector quarkonia in terms of the total
angular momentum of quarks and gluons. Specifically, we find the spin contents
of $J/\psi$, $\chi_{c1}$, $\Upsilon(1S)$, and $\chi_{b1}$ are slightly
different from the naive quark model picture. For example, the $J/\psi$ is
traditionally considered as an S-wave particle, but we find quarks do not carry
all of the total spin.
