Partitions of unity in $SL(2,\mathbb Z)$, negative continued fractions and dissections of polygons

We characterize sequences of positive integers $(a_1, a_2,\dots, a_n)$ for which the $2\times 2$ matrix given by the product of the elementary matrices $\left(
\begin{array}{cc}a_j & -1 \1 & 0 \end{array}
\right)$is either the identity matrix Id, its negative ? Id, or square root of ? Id. This extends a theorem of Conway and Coxeter that classifies such solutions subject to a total positivity restriction