  Finite-size scaling at fixed renormalization-group invariant is a powerful
and flexible technique to analyze Monte Carlo data at a critical point. It
consists in fixing a given renormalization-group invariant quantity to a given
value, thereby trading its statistical fluctuations with those of a parameter
driving the transition. One remarkable feature is the observed significant
improvement of statistical accuracy of various quantities, as compared to a
standard analysis. We review the method, discussing in detail its
implementation, the error analysis, and a previously introduced
covariance-based optimization. Comprehensive benchmarks on the Ising model in
two and three dimensions show large gains in the statistical accuracy, which
are due to cross-correlations between observables. As an application, we
compute an accurate estimate of the inverse critical temperature of the
improved O(2) $\phi^4$ model on a three-dimensional cubic lattice.
