Since Mr. Romney didn’t use all the foreign tax credit available to him in 2009, tax experts said he must have reduced the tax owed on his foreign income to zero. In the highly unlikely event he also reduced his taxable income from United States sources to zero, or even showed a loss, he would have owed no federal income tax in 2009.

That’s not the only evidence that Mr. Romney’s income in 2009 was substantially lower than in 2010. He was able to use substantial losses from previous years to reduce his 2010 capital gains, indicating that he had no net capital gain in 2009. Capital gains accounted for most of Mr. Romney’s income in 2010 — $12.6 million of his $21.6 million total. In addition, he reported more than $800,000 in taxable refunds from 2009, which seems very high. If he had smaller refunds, lower speaking and director’s fees, and lower business income or even a business loss, his adjusted gross income for 2009 could have been quite low, at least by Mr. Romney’s standards. Assuming he had similar itemized deductions to the $4.5 million he had in 2010, his taxable income could have been extremely low.

The data Mr. Romney filed also indicated that his foreign tax liability shot up in 2008, the year the financial crisis began. His total foreign taxes amounted to nearly $800,000, far more than any other year. Even if his foreign tax rate was the comparatively high rate that prevails in the United States, 15 percent, that suggests substantial income, more than $5 million from his foreign assets.

How could Mr. Romney’s blind trust have earned such amounts in foreign countries and generated such large tax credits? One possibility is that the financial crisis and resulting bleak economic outlook caused Mr. Romney’s trustee to cash in long-held, appreciated foreign assets. Another possibility is that they resulted from some kind of more elaborate tax shelter, or some combination of the two. The issue has baffled tax experts.

The big foreign tax credit that year “makes no sense to me at all,” Professor Shaviro said. “We have no idea what he actually did.” But if it is the result of some kind of tax shelter, “The key in general is to try to exploit technical gaps in the rules by using transactions that have little or no economic substance.” He said the I.R.S. had challenged many such deals intended to generate foreign tax credits.

Professor Hines added: “Lots of people are speculating about the source of this tax credit, but at this point it’s just conjecture. Why doesn’t he just release the returns?”

Mr. Romney doesn’t even have to release his 2008 and 2009 returns to answer questions about numbers he has already made public on his 2010 returns. Since his foreign income and resulting tax treatment came through his blind trust, he presumably doesn’t know the answers, but his trustee would. If there’s a simple explanation — or any explanation for that matter — no one is offering it. After I discussed the foreign tax credit issues with a spokeswoman for the campaign, she declined to comment.

Mr. Romney aside, the complexity and frequent abuse of the foreign tax credit provisions seem to make another compelling case for tax reform. Professor Shaviro is writing a book on the topic, and says the foreign tax credit as it now exists is a “nightmare,” adding: “It’s subject to abuse. If it was up to me, I’d wipe out the credit, wipe out the deferrals, and simply tax all foreign income at a single low rate.”