By Kevin Drawbaugh WASHINGTON (Reuters) - The U.S. government has grappled for more than 30 years with corporate deals known as inversions in which U.S. companies shift their tax domiciles abroad to avoid U.S. taxes. Fifty-two substantial deals like this have occurred since 1983, about half of them since the 2008-2009 credit crisis, according to a Reuters analysis. Here is a summary of U.S. government efforts to manage them. ISLAND TIME The first inversion in 1983 was followed a year later by two new rules from the U.S. Internal Revenue Service. No more big inversions followed for more than a decade. In the mid-1990s, the deals resumed, typically with U.S. corporations setting up smaller companies in Bermuda or the Cayman Islands. Many energy and insurance companies did this. In these inversions, the new island tax haven company would become the parent of the U.S. business, though core operations stayed in the United States. The name "inversion" came from the idea of turning the company upside down, making the offshore company the head and U.S. operations the body. Inversions allowed U.S. companies to book foreign profits at low or no taxes abroad, putting them out of the reach of the U.S. Internal Revenue Service. By booking the income in a foreign country with a lower tax rate, those profits could quickly be put to use at a lower cost. Plus, the strategy made other legal tax-avoidance moves easier, such as U.S. earnings stripping. That involves a foreign parent lending to a U.S. unit, which deducts interest payments to make its U.S. income smaller for tax purposes, while the foreign parent books interest income at its home country's lower tax rate. TREASURY RESPONDS The U.S. Treasury Department expressed concern about the deals in 2002. Congress in 2004 adopted Section 7874 of the tax code. It set two tests of whether an inverted company would be treated as foreign or domestic by the U.S. government. First, if the original U.S. company's investors still held 80 percent or more of the new foreign parent's shares, the new parent would be treated as a U.S. company, not a foreign one. Second, the same treatment would apply if the new, foreign parent had no "substantial business activities" in its home country. This specifically targeted island tax-haven holding companies that were often little more than a post office box. After these rules were adopted, the deals dried up again. NEXT WAVE Another wave started in 2008, but this time, the deals had changed. Most no longer targeted island havens. Instead, new foreign parents for inversions were being acquired in buyouts and mergers in Canada, Ireland, Britain and the Netherlands. Also, deals were being structured to take advantage of a part of the law that said if the original U.S. shareholders owned from 60 to 80 percent of the new, foreign parent, it would be treated as a foreign entity, but with some restrictions. The IRS issued new rules, but they were repeatedly revised. OBAMA'S PROPOSALS By 2010, large U.S. businesses were pursuing inversions. President Barack Obama asked Congress to act on proposals from the White House to curb the deals. Over the past four years, Obama has annually called for one or more of the following changes. One is to make inversions harder to do by scrapping the 60-80 percent test and changing the 80 percent original investors' test to 50 percent. The president has urged a "substantial business activities" test that would deny foreign company status where operations are still primarily located in the United States and U.S.-managed. He has called for extending inversion limits to partnerships. Finally, Obama has repeatedly urged fighting U.S. earnings stripping by tightening the limits on deductibility of interest paid by inverted U.S. companies on inter-company debt. While several bills have been filed by Democratic lawmakers, Congress has taken no action on inversions in many years. (Reporting by Kevin Drawbaugh; Editing by Martin Howell)