Are you wanting to view this from a modern point of view, or at the time? Obviously something like a political spectrum is ever-shifting: in present day United States we wouldn't say that the "right" are monarchists for example. I'll cover it from the point of the of the French since I think that's the most beneficial way to view it.
FUN FACT ALERT! The "right" and "left" political designations we still use today originated during the French Revolution! The deputies of the National Assembly (formerly the Estates General) would commonly sit with those who shared their political beliefs. The conservatives would sit on the right end of the hall, while the liberals preferred the left. Thus they began to refer to each other in these spatial terms, i.e. "oh no the left is proposing another bill on church lands today" and the designation stuck!
So in discussing the French Revolution it's always crucial to keep in mind that there was not one big 'burst' of revolution, and that it was an evolving and changing process, especially from 1789-1794. Therefore the answer isn't static, but changing over time. Let's go through some of the main changes in power dynamic between right and left. I'll keep it at a high level here, but just know that the power struggle was constantly evolving.
- Start of the Estates General - 1790: Right off the bat the revolution was driven by 'liberal nobles'. These men were very well off, but were upset with the present order for various reasons. They believed that they should be taxed equally with the rest of the population. These men began to lose power after the move from Versailles to Paris.
- 1790-1791: The liberal-er liberal nobles. After the more conservative liberal nobles were alienated (mostly due to questions of the church and the make-up of the constitutional monarchy), the nobles of the "Society of 1789" stepped up to the plate. These men were 'enlightened' on the question of religion, and could be said to constitute a middle position between right and left.
- 1791-1792: Things get spicy here. The Feuillants, a group slightly to the left of the Society of 1789, has taken control under the Legislative Assembly, which was the representative gov't called under the newly minted Constitution of 1791. The heat gets turned up because Louis gets cold feet and flees Paris in the ill-fated Flight to Varennes. After he's 'returned' by his people things take a serious shift to the left. I personally agree with Historian Timothy Tackett's point of view that this is THE moment when the Revolution goes from moderate and measured to the chaotic event we know and love it for.
- 1792-1793: It's Republic time! Important to know: no one was advocating for a Republic right up until around mid-1792. It was entirely unthought of in the early Revolution. We kind of have this concept that the revolutionaries were always pushing for a republic as the best form of government, but that definitely wasn't the case. Even the most radical of the radical like Marat were perfectly happy with a Constitutional Monarchy until Louis started showing what kind of a monarch he'd be. After the Insurrection of August 10th, we go from 'moderate liberalism' to a liberal republic, based upon universal manhood suffrage.
- 1793-1794: The most radical phase of the Revolution. This is a time many have tried to identify as 'proto-communist'. The Revolutionaries also played around with concepts like the right to existence, price controls on foodstuffs. The Constitution of 1793 is considered to be one of, if not the most, liberal constitution ever written.
- Thermidor - Napoleon: After the fall of Robespierre things will gradually creep back towards the right into a sort of moderate republicanism, until Napoleon takes power. Not nearly as fun after 1794, so I'll end here.