For some reason I have been under the impression that US law prohibits
US citizens from retaining their citizenship if they ever obtain a new
citizenship. Is this always, sometimes, or never true? If sometimes,
Nope. The U.S. Supreme Court ruled in Afroyim v. Rusk in 1967 that someone who was born or naturalized in the U.S. was constitutionally protected from losing their U.S. citizenship involuntarily, "involuntarily" as in without their intent to do so. In Vance v. Terrazas, the Supreme Court ruled that this intention must be proven with "preponderance of evidence" -- Congress cannot set a lesser standard for the intention (e.g. Congress cannot say naturalizing in a foreign country is enough to show intention to give up U.S. citizenship).
Note that this constitutional protection does not apply to all U.S. citizens -- people who were not born or naturalized in the U.S. (e.g. people born abroad but were citizens from birth, derived through parents) are not protected by the 14th amendment, and Congress can take away their citizenship without their intention. (Supreme Court affirmed this in Rogers v. Bellei.) In the past, there have been "retention requirements" for U.S. citizens from birth abroad, that if not met would mean the person would lose their U.S. citizenship at a certain age. However, no such law exists now. So, as of right now, all U.S. citizens cannot lose their U.S. citizenship without their intention to do so, no matter what they do, including getting another citizenship.
Some people have had their U.S. citizenship taken away involuntarily prior to 1967 for things like naturalizing in a foreign country and other things, but I believe that if it were to come up in court today it would be found that they did not really lose U.S. citizenship.
8 USC 1481 is the current law for loss of U.S. nationality. It requires the performance of one of several "potentially expatriating acts", combined with intention to relinquish U.S. nationality, to lose U.S. nationality. One of the potentially expatriating acts is voluntarily obtaining naturalization in a foreign country.
Curent State Department policy makes it even harder to lose U.S. citizenship this way. With voluntary naturalization in a foreign country (and a few other of the potentially expatriating acts), the State Department will automatically presume that you do not intend to relinquish U.S. nationality. If you want to keep U.S. nationality, you don't need to do anything special. If asked whether you want to give up U.S. nationality, you can say no, and you will not lose U.S. nationality.
Or perhaps I have the situation reversed in my head, and US law
requires new citizens to renounce their previous citizenship when
acquiring US citizenship?
This is also not the case.