Much of the left vs right divide is more like a revolving door where age and leadership character seem to define the politics of the times. In the current era most people identify religion with conservative, authoritarian, reactionary politics but many times in the past religion has functioned as the progressive political front provoking reactionary authoritative governments to clamp down against religion. Most of the time rulers co-opt the support of the religious authorities for a veneer of legitimacy.
As for leftward or rightward being more or less controlling... plenty of examples on both sides. Most of the time ideology has little bearing on what actually happens. The reality is more often that certain people are unhappy with the status quo and adopt whichever politics of the moment seems best fit to rocking the boat and giving them what they would otherwise have difficulty attaining under the normal situation. Very few revolutions are won by those who began them- usually the first wave of revolutionaries are struck down by the system they are revolting against and the 2nd wave uses their martyrdom to further the revolt but eventually succumb to the excesses of the revolution and it is the men leading the military or guardians of purity (secret police, many names can be adopted for this role but the political claim is usually about protecting the purity of the revolution even if the results bear nothing close to what the original revolutionaries called for).