The crisis facing Israel in the West Bank and Gaza today is the loss of face, not territory. Israel has built its international persona as the feisty underdog, surrounded by belligerent enemies. But since December, the world has watched that underdog shoot live ammunition at young people armed with rocks.

Hard-liners in Israel are calling for more repression of the Palestinian upstarts, while Palestinian religious leaders are more than happy to fuel their cause with martyrs. The policies of each side provide momentum for more violence and more killing.

Our White House sources tell us that President Reagan has been furious about the way the Israelis have handled the situation in the West Bank and Gaza. Reagan has surprised the Israelis with his strong sympathies for the Palestinians. The same sources tell us that Reagan has not forgotten how the Israelis scuttled his well-thought-out peace plan in 1983.

That was a time when Reagan made a strong pitch on behalf of the Palestinians to then-Israeli Prime Minister Yitzhak Shamir. We have seen a confidential briefing paper that says Reagan felt the main obstacle to a Middle East peace was Israel's policy of encouraging Jews to settle on the West Bank. He asked for a freeze on Jewish settlements. Shamir vaguely agreed to cooperate with the State Department, but there never was any real cooperation.

Reagan knows Congress is still very susceptible to the Israeli lobby. Sources report a sense of ''helplessness'' in the White House whenever Reagan tries to get Congress to influence Israeli action in the occupied territories.

Congress killed a $23 million proposal to improve living conditions for Palestinians in the territories. And new financing procedures were written into the budget bill at the last minute that will save Israel $2 billion in paying off its huge debt accumulated over years of American aid.

Pro-Israeli pressure from Congress was so strong as the violence began in December that Assistant Secretary of State for Near Eastern Affairs Richard Murphy was ordered to reassure the Israeli ambassador to Washington that there would be no change in U.S. commitment to Israel, despite the administration's stance.

The Israelis are deliberately blaming the wrong party for the violence. They say the Palestine Liberation Organization is behind it. The PLO wants the credit, but the real force behind the riots is a younger generation of Palestinians who have rejected the PLO as too moderate. This new group takes its cues from more radical sources, such as terrorist Abu Nidal's organization and a new Palestinian Sunni fundamentalist movement encouraged by Iran.

Sources on the West Bank tell us the young Palestinians are no longer afraid of the Israelis. One American expert on this issue, Dr. Omar Kader, told us that ''this generation of Palestinian youth is no longer content to follow moderate leaders who have been ignored by the Israelis and the United States. They now believe only radical, even violent, methods will be taken seriously by the Israelis.''

Israel's heavy-handed reaction to the demonstrators is playing into the hands of these radicals. By punishing every citizen for the actions of a few, Israel is stirring up a fresh supply of bitter enemies.

One West Bank source told us that before the demonstrations, only young radicals talked of challenging the Israelis. ''Now everyone considers it their duty to risk their lives to end the occupation of the territories by Israel.''