How to evaluate models: Observed vs. predicted or predicted vs. observed?

A common and simple approach to evaluate models is to regress predicted vs. observed values (or vice versa) and compare slope and intercept parameters against the 1:1 line. However, based on a review of the literature it seems to be no consensus on which variable (predicted or observed) should be placed in each axis. Although some researchers think that it is identical, probably because r2 is the same for both regressions, the intercept and the slope of each regression differ and, in turn, may change the result of the model evaluation. We present mathematical evidence showing that the regression of predicted (in the y-axis) vs. observed data (in the x-axis) (PO) to evaluate models is incorrect and should lead to an erroneous estimate of the slope and intercept. In other words, a spurious effect is added to the regression parameters when regressing PO values and comparing them against the 1:1 line. Observed (in the y-axis) vs. predicted (in the x-axis) (OP) regressions should be used instead. We also show in an example from the literature that both approaches produce significantly different results that may change the conclusions of the model evaluation.