They were once bitter enemies, and they are still far from being close friends.
Nevertheless, Jordan and the Palestine Liberation Organization are steadily progressing toward reconciliation. The going is quite slow - and that means a diplomatic breakthrough in the Middle East peace process is a long way off.
As they meet to improve relations, both Jordanian and PLO leaders are conscientious about issuing and reissuing unequivocal statements of assurance to Arab hard-liners - meaning, primarily, the PLO rank and file and Syria. Jordan is not trying to usurp the PLO's role and make a separate, Egyptian-style peace with Israel, the leaders say, and the PLO is not selling itself out to Jordan.
PLO chairman Yasser Arafat met in Amman this week with Jordanian officials, the third time Mr. Arafat has been in the Jordanian capital since the PLO evacuated Beirut in August. His discussions there concerned strategy coordination, and in particular, agreement on a joint PLO-Jordanian position to be presented by Jordan's King Hussein to President Reagan when the monarch visits Washington Dec. 21. The Jordanian delegation is expected to include non-PLO Palestinians.
The vehicle for policy coordination is the Jordanian-Palestinian Joint Committee, which links Jordan and the PLO with the West Bank and serves as a conduit for Arab financial aid to Palestinians under occupation. Significantly, Mr. Arafat was received in Amman Dec. 13 by Jordan's minister for affairs in the occupied territories, Hasan Ibrahim.
In the longer run, Jordanian-PLO coordination meetings are trying to come up with a bilateral position on the composition of a future Palestinian state and its links to Jordan.
But because the reconciliation process is delicate, it will take many more months - and many more meetings - before Mr. Arafat and King Hussein can possibly feel confident enough to strike off in a bold diplomatic direction. If rapprochement continues unhindered, however, it is conceivable that one day the PLO and Jordan could form a joint negotiating team to discuss the West Bank and Gaza with the US and/or Israel.
At the moment, that prospect is not apparent. Mr. Arafat and Jordanian officials are busy assuring Arab hard-liners that there is no change in the status quo in which the PLO is the sole party authorized to talk about turning the West Bank, Gaza, and East Jerusalem into an independent Palestinian state under PLO leadership.
The latest issue of the PLO weekly magazine Falestine Al-Thawra told its readers that relations between Jordan and the PLO are being improved gradually ''based on the belief that unity between Jordan and Palestine cannot be established as long as a Palestinian state does not exist.''
Jordanian Information Minister Adnan Abu Odeh made it clear Dec. 12 that Jordan still considers the PLO the sole representative of the Palestinian people. He said no one should expect either the PLO or the Palestinians in general to be excluded from negotiations concerning their future.
These statements are designed to preempt critics - especially critics based in Syria - who are concerned that Jordan wants to replace the PLO in negotiations with Israel and incorporate the West Bank into the Kingdom of Jordan.
Jordanians and Palestinians remember the costly fighting in Jordan in 1970 and the ultimate expulsion of the PLO by King Hussein. PLO suspicions of Hussein also are rooted in the King's 1972 plan to create a United Arab Kingdom out of Jordan and the West Bank and Jordan's reticence to aid the PLO in times of trouble - especially during last summer's Israeli invasion of Lebanon.
PLO moderates have to be wary of Syria. Most of Mr. Arafat's fighters are effectively under President Hafez Assad's control in Syria or in Lebanon.