A sequence $(P_n(x))$ of polynomials with $\deg(P_n(x)) = n$ is {\it orthogonal} with respect to a linear functional $L$ if $L(P_m(x) P_n(x))$ is nonzero if and only if $m = n$. Such a family is intrinsically linked to a recurrence relation $$ P_{ n+1 } (x) = (x-b_n) P_n(x) - \lambda_n P_{ n-1 } (x) $$ for some sequences $(b_n)$ and $(\lambda_n)$, with initial conditions $P_{ -1 }(x) = 0$ and $P_0(x) = 1$. The $n$th {\it moment} of this sequence of orthogonal polynomials is defined to be $L(x^n)$. Classic results by G. Viennot [``Une theorie combinatoire des polynomes orthogonaux generaux'', lecture notes, Univ. Québec Montréal, 1983] express the moments of orthogonal polynomials in terms of Motzkin paths. \par A variation of these moments, where the Motzkin paths satisfy certain bound conditions, exhibits a recurrence relation (in $n$), which can thus be run backwards, hence the name {\it negative moments}. Recent work of J. Cigler and C. Krattenthaler [``Bounded Dyck paths, bounded alternating sequences, orthogonal polynomials, and reciprocity'', Eur. J. Comb., posted October 9, 2023, \doi{10.1016/j.ejc.2023.103840}] gave an interpretation of negative moments for $b_n = 0$ and $\lambda_n = 1$ in terms of alternating sequences; this is an example of a combinatorial reciprocity theorem. \par The paper under review gives a novel proof of the Cigler-Krattenthaler theorem using continued fractions. The authors extend this to a more general reciprocity theorem and prove two conjectures of Cigler and Krattenthaler.