Dog whistle politics
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The term alludes to the sound of a dog whistle, which can only be heard by the intended audience (the dog). In theory at least, dog whistle terms are only noticed and understood by the people they are intended for.
 States' rights
The best known example of this is the love some older conservatives have for "states' rights." When segregation became socially unacceptable, many old pro-segregationists began instead calling for "states' rights," meaning of course the right of the individual states to segregate and discriminate (or, in the pre-Civil War American South, enforce slavery). Since then, many civil rights leaders have been able to show the connections behind the phrase, but that doesn't stop hard-right Ron/Rand Paul supporters from continuing to use it. These days, it's used by people in red states to justify tighter immigration law (i.e. racial profiling) and toying with Roe v. Wade by restricting abortions.
 Religious, class, and race-baiting
This is a common way these days of expressing bigotry without appearing to do so. Anti-Semitism is often expressed in dog whistle terms, which are picked up by those familiar with the main concepts of anti-Semitism but not by the populace as a whole. Examples are talk about the "controlled" media (without specifying who controls it, but anti-Semites know "who" that means); "international bankers" (without specifying Jewish international bankers, but anti-Semites know who); etc. "Dual loyalty" is a dog whistle term used by both anti-Semites and anti-Catholics to imply the person's real loyalty is to Israel or the Vatican and not their own country. Another term is anti-Zionism, or criticism of Israel. While people can legitimately criticize Israel for a great many reasons, professed anti-Zionism is sometimes a thinly-veiled front for antisemitism.
Another way to complain about minorities without offense is to complain about ghettos, the bad part of town, etc. Or if one's a politician, "fixing" the impoverished areas of a city will do the trick. Particularly in the Midwest, it's common for people to mock places like Detroit, Cleveland, and the South Side of Chicago as it sounds innocent to the untrained ear.
In addition, since the civil rights movement rendered open racism in the United States anathema, dog-whistle terms such as "community organizers" and "welfare queen" have become widely known as semi-opaque terms referring to people (usually African-American, always underclass) who propagandists want you to believe are stirring up trouble and abusing the system. "Quotas" is a semi-opaque way of signaling opposition to Affirmative Action or other efforts to achieve gender or racial parity (in areas such as highest educational attainment, proportional political representation, wages, etc.) without coming right out and directly saying it.
For homophobes, pointing out that the opposing political candidate is "single," "never married", "a lifelong bachelor," limp-wristed or even "has no children" if they are married, is a dog whistle term intended to send a signal to homophobes that others will miss. Similarly, candidates who wish to let their voters know that they are against gay rights will often say they support "family values".
This sort of dog-whistle talk used to be a lot more common regarding matters of sexuality in general prior to the sexual revolution, as in for example various euphemisms used in place of openly saying that a single woman was pregnant or had had an abortion.
 Abortion foes
A less well-known example is when pro-life activists talk about the "Dred Scott Decision." Among them, Roe v. Wade is analogous to Dred Scott v. Sandford, in that it keeps fetuses from having any rights. However, since Roe v. Wade is popular, and Dred Scott is not, a politician (or potential Supreme Court Justice) can talk about Dred Scott when they are really talking about Roe v. Wade.
 Fringe Christian sects
"Anointed" is a dog whistle term for speaking in tongues, name it and claim it, faith healing, and other fringe practices, intended to be understood by Pentecostals but not by the general public. It is a way of saying "we believe in and practice the Pentecostal beliefs" without openly admitting it.