Let $n\ge 1$, and let $(M^{2n+1},T^{1,0}M)$ be a compact smooth CR manifold, where $T^{1,0}M\subset \mathbb C TM$ is the rank-$n$ CR bundle of complex tangent directions. Let $T^*M$ denote the real cotangent bundle of $M$, and let $\theta\in C^\infty(M;T^*M)$ be a pseudohermitian contact form compatible with this CR structure. Let $\mu_\theta$ denote the smooth measure induced by the pseudohermitian volume form $\theta\wedge(d\theta)^n$, and let $Q_\theta\in C^\infty(M;\mathbb R)$ be the CR Q-curvature associated to $\theta$. Suppose that, for every smooth real-valued function $\Upsilon\in C^\infty(M;\mathbb R)$, the contact form $\widehat\theta:=e^\Upsilon\theta$ has associated CR Q-curvature $Q_{\widehat\theta}\in C^\infty(M;\mathbb R)$ and associated pseudohermitian measure $\mu_{\widehat\theta}$ induced by $\widehat\theta\wedge(d\widehat\theta)^n$. Suppose further that there is a real linear differential operator $P_\theta:C^\infty(M;\mathbb R)\to C^\infty(M;\mathbb R)$ which is formally self-adjoint with respect to $\mu_\theta$, satisfies $P_\theta 1=0$, and obeys the CR Q-curvature transformation law $e^{(n+1)\Upsilon}Q_{\widehat\theta}=Q_\theta+P_\theta\Upsilon$ for every $\Upsilon\in C^\infty(M;\mathbb R)$. Then the total CR Q-curvature $\int_M Q_\theta\,d\mu_\theta(x)$ is independent of the choice of contact form in the pseudohermitian conformal class $\{e^\Upsilon\theta:\Upsilon\in C^\infty(M;\mathbb R)\}$.