If I live in Maine but work in New Hampshire, do I owe income tax in Maine on my New Hampshire-based income?
I do not believe that Maine and New Hampshire have any tax reciprocity agreement.
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If you are a resident of Maine, you owe taxes on all of your taxable income, regardless of where you work. Since New Hampshire has no personal income tax, there is no credit for (hypothetical) taxes paid to New Hampshire on any portion of that income.
From Determining Residency Status (pdf):
If you are a Maine resident for the entire tax year, you must pay Maine tax on all of your taxable income regardless of its sources – wages, investment income, interest income, pension, and dividends among other things....
If you are subject to income tax by another state or similar jurisdiction in another country on some of your Maine taxable income, you may be allowed a credit against Maine income tax for all or some of the tax paid to the other state or jurisdiction.
You're correct, Maine and New Hampshire do not have a tax reciprocity agreement. The usual course of action is to have taxes withheld at your working location state and then you will have to sort it out when you file in both New Hampshire and Maine.
Usually you will owe taxes in your state of residence and be owed a refund in your work state but situations vary.
You may want to consult a tax accountant who is familiar with ME and NH tax laws and be sure you are following the most advantageous course of action.