Is the definition of Hazard Ratio strict? Is it always (even if only by convention) hazard rate in intervention group divided by hazard rate in control group?
 A: Hazard ratios apply to many more situations than control/intervention groups. If you have a continuous predictor like age or blood pressure, the hazard ratio in a Cox regression would be the change in hazard per unit change in age or pressure. If you have a categorical predictor (say, multiple subtypes of a disease as in breast cancer, or gender) then you arbitrarily choose one category as a reference against which to compare the others.
In the control/intervention context it's certainly usual to choose the control as the reference group. I am, however, aware of no law requiring that. In some contexts you might choose to switch reference groups or the direction of a continuous variable (e.g., hazard per year decrease rather than per year increase in age). For example, when you examine a plot of the relative entries of predictors as the penalty changes in a LASSO model it's most convenient (at least for me) for the predictors to have either all HR >1 or all <1, obtained with appropriate choices of reference levels or directions, so that all the curves start by pointing all up or all down when they leave the horizontal axis.
If you are reading, pay careful attention to the authors' choice of reference level. If you are writing, choose reference levels that make it easiest to explain to your audience.
