Lawrence O'Donnell on Presidential tax returns.

Democrats are hoping that President Trump’s federal tax returns will serve as a Rosetta stone to unlock the secrets of his murky finances. Mr. Trump, on the other hand, has spent the past three years refusing to divulge the returns and even dismissing them as boring.

Now, the fight over whether members of Congress have the right to see the president’s tax returns is intensifying. The House tax-writing committee has demanded them, and late Wednesday, Treasury Secretary Steven Mnuchin said his department would not comply with the request by the date set by the committee. Earlier in the day, Mr. Trump repeated that he would not disclose his returns, falsely claiming that there is no law that gives Congress the authority to obtain them.

Mr. Trump is the first president in four decades to not publicly release his tax returns. The refusal — coupled with his continued ownership of a far-flung network of closely held businesses — has fanned suspicions that he has something to hide.

So, what might lawmakers find if they obtain Mr. Trump’s tax filings?

For starters, his recent returns could illuminate the sources of his income and any tax-avoidance strategies he may have used, but it would not fully decode the family’s personal and business finances.

Leaked pages of the president’s old tax returns and other financial documents already have shed light on his finances. They indicate that Mr. Trump most likely avoided paying income taxes for a number of years and that in other periods The New York Times found he participated in fraudulent tax schemes to minimize his tax bill. His federal ethics filings show hundreds of millions of dollars in outstanding loans and a business network owned through a web of corporate entities.

The House Ways and Means Committee last week requested six years of Mr. Trump’s personal tax returns, along with the returns filed by one of his trusts and seven subsidiaries he controls. They also asked for records of any audits.

The returns would partly reveal the sources of the president’s income, and whether his businesses are profitable. They would show how much, if anything, Mr. Trump has been paying in taxes and might show whether he has been aggressive in reducing his tax bills. They will show whether Mr. Trump has personally donated to charity.

The returns requested by the committee, however, would not fully answer longstanding questions about the identities of Mr. Trump’s business partners, the ultimate sources of his income or whether he is entangled with entities that have interests before his administration.

Even if Representative Richard E. Neal, Democrat of Massachusetts, who leads the committee, is successful, it does not mean the tax returns will become public. By law, they are supposed to remain solely in the hands of the committee; an unauthorized release would be a felony.

How much is Mr. Trump worth?

Ever since he declared his candidacy, Mr. Trump — a self-proclaimed billionaire — has faced questions about whether he is exaggerating his wealth.

Mr. Trump has never substantiated his assertion that he is a billionaire, though he has repeatedly made the claim to banks and other institutions. The disclosure of his tax returns probably would not settle the debate over Mr. Trump’s net worth.

Tax returns do not provide lists of assets or liabilities, unless they generate income or deductions. Even then, the returns most likely would not provide values of those assets or liabilities.

Still, Mr. Trump’s full tax returns — his personal returns and the ones for all of the components of the Trump Organization — would give a sense of the profitability of his businesses.

The returns would also reveal whether his companies have taken a hit since he entered the White House. (Mr. Trump has said he has lost vast amounts of money since he became president.)

Whom is Mr. Trump in business with?

Mr. Trump for years ran the Trump Organization, a privately held company that manages real estate, golf resorts, brand and licensing deals and other interests. Much about the businesses’ finances remains unknown.

The federal ethics disclosures Mr. Trump has made about his holdings so far have been limited. For example, there is no requirement that he report the identities of investors who have put money into his businesses.

Mr. Trump acquired many properties, in particular golf courses, with cash, but where did that cash come from? Mr. Trump’s son Eric once told a journalist that the Trump Organization’s golf properties were paid for with Russian money. Eric Trump has denied making that statement.

While Mr. Trump’s federal filings have reported some outstanding loans, investments and income from each business, they do not include debts where he has a partial interest or older loans that have been paid off. In theory, the existence of those older loans — and the lenders behind them — could be disclosed on his historical tax returns because certain types of interest on debts can be tax-deductible.

In reality, much of that information is unlikely to be contained in the limited number of returns requested by Congress. The panel is seeking the returns for only a small number of Mr. Trump’s companies. But he controls hundreds of separate L.L.C.s that contain individual businesses, as varied as a particular building or the rights to license his name overseas. Each one of those businesses may file its own tax return, separate from the ones requested by the committee.

How exactly does Mr. Trump make his money?

Mr. Trump owns and operates businesses in the United States and overseas. He earns revenue from golf club memberships, from leasing space in his buildings and from licensing deals to put the “Trump” name on other developers’ properties.

Mr. Trump’s government disclosure reports show ranges of how much revenue his various subsidiaries collect, as provided by Mr. Trump, but they do not indicate the companies’ overall profitability.

For example, his most recent government ethics report states that his Irish golf business reported revenue of $14 million in 2017. However, Irish regulators require that the golf subsidiary make its filings public. Those disclosures show the business actually lost about $2 million that year. Mr. Trump’s returns could provide further detail about the profitability of his other businesses. More broadly, his tax returns can be compared with the ethics disclosures to check the accuracy of the filings.

Mr. Trump’s personal tax returns will also itemize his stock- and bond-market transactions and people and institutions he is receiving income from, domestic and foreign. This would include income he has received from banks.

Is Mr. Trump cheating on his taxes?

If Democrats get Mr. Trump’s taxes, they will be looking for any aggressive or illegal maneuvers — such as the fraudulent tactics he has used in the past.

Sophisticated tax filers use their returns to paint the picture of their finances they want the government to see. It usually takes a lot of digging to find out what is going on behind the numbers.

For example, as The New York Times reported last year, Mr. Trump and his siblings set up a sham corporation in the early 1990s to disguise millions of dollars in gifts from their parents.

In that case, The Times had access to far more than just tax returns: It also relied on confidential business filings, public records and information from sources to document the fraud.

At the time the article was published, a lawyer for the president, Charles J. Harder, said the allegations of fraud and tax evasion were 100 percent false, and highly defamatory.

What will be more immediately apparent in Mr. Trump’s tax returns is what sort of deductions he has used to reduce his tax bill, and how aggressively he has used them.

In 1995, for example, Mr. Trump paid no income tax, and records show he has not paid income tax in several other years.

The president has said he has avoided taxes because of the generous deductions given to real estate owners like himself. One deduction is depreciation, which permits real estate investors to deduct a portion of the cost of their buildings from their taxable income.

“I have a write-off. A lot of it is depreciation, which is a wonderful charge,” he said during a presidential debate in 2016. “I love depreciation.”