Insinuation and innuendo. Unwise use of this word may seriously damage your linguistic credibility. It is one of the few words in the Second Edition of the Oxford English Dictionary to which an editorial note has been attached: "A tasteless word," it says, signed with the initials of the editor.
It is an obvious blend of insinuation and innuendo, a portmanteau construction containing elements of both. Throughout its life (and it is more than a century old, as we shall see), it has been used either humorously, or to indicate that a speaker in a book or play is uneducated.
The OED knows of it from 1885. Its first example says that it was invented by a legislator from South Carolina. This seems to be confirmed by this little squib, which appeared in Appletons' Journal of New York in 1875. It manages the trick of being both heavy-handed and tongue-in-cheek at the same time:
The South Carolina Legislature has immortalized itself by coining the word "insinuendos." Seeing the wideness of its application, the Tribune begs to be "permitted to express the obligations which society, and especially society's representatives in official life, legislators, cabinet officers, and such, are under for an uncommonly fresh, beautiful, and expressive phrase. It admirably fits the time. It is a contribution to current politics as well as to philology."
I can reveal that this belief about its origin is wrong. The unnamed legislator from South Carolina may have re-invented the word, but it was around earlier, since it appeared in 1871 in a now totally forgotten one-act comic drama by William R Emerson called Putkins. In this, he has an uneducated person say "I scorn the insinuendo!" Whether Mr. Emerson invented it, or whether he borrowed a term already known, the linguistic record is as yet unable to say.
It is rarely seen today, but anyone using it would have to flag their facetious intent, or be assumed to be as ignorant as Mr Emerson's character. But it is inoffensive enough, in all truth, and neatly encapsulates two ideas themselves closely related.