PARIS (Reuters) - A very fragmented French parliament meets on Thursday for the first time since an inconclusive snap election to appoint the assembly's president, in a ballot that could be key to who calls the shots in French politics for years to come.

No party won an absolute majority in the election earlier this month, in which the left unexpectedly came first, ahead of President Emmanuel Macron's centrists and Marine Le Pen's far right.

Electing the lower house of parliament's president, equivalent to a speaker who organises the chamber's agenda and runs debates, is usually a formality.

But it is very different this time.

With Macron weakened and uncertainty over who will form the next government and how effective it could be in a divided parliament, who runs the assembly will be key.

The New Popular Front (NFP) left-wing alliance, hastily assembled before the snap election, wants to run the government, but has been fighting bitterly since then over who to put forward as prime minister.

It hopes its candidate for parliament chief, veteran Communist lawmaker André Chassaigne, who is well liked across party lines, could be elected. This could be a boost to its chances of forming a government, by showing the alliance can command a majority in the assembly.

Chassaigne's election would show parliament's "freedom and independence from the executive," Benjamin Lucas, a Green lawmaker from the NFP group told Reuters.

But the outcome is far from certain. Macron's camp could be striking a deal with the conservatives to try to get the outgoing parliament chief, Yaël Braun-Pivet, who belongs to Macron's Together group, re-elected.

They in turn hope that would put mainstream parties in a stronger position to forge an alliance to form a government that could include some of the NFP but exclude the hard-left France Unbowed.

The far-right National Rally (RN), though it has little hope of getting the speaker role, is vying for other key roles in the assembly, including chairing the powerful finance committee.

Other candidates for parliament president include centrist Charles de Courson - who does not belong to Macron's party - and the RN's Sébastien Chenu.

The vote will take place in three rounds, starting at 3 p.m. (1300 GMT) and alliances are bound to be struck off between rounds, with some candidates pulling out.

Votes are confidential. To win in the first round, a candidate would need an absolute majority of votes cast, which is unlikely to happen. The process is the same for the second round. In a third round, the candidate with most votes wins.

(Reporting by Ingrid Melander and Elizabeth Pineau; Editing by Ros Russell)