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Question:

Is there any evidence that any of the world's largest airlines¹ (e.g. American Airlines, Delta, United) reuse the blankets that are often handed out on international flights, with or without cleaning them? Inspired by Took airline plane blanket by accident; didn't realize it was forbidden. What should I do?

Prior research:

Googling has turned up a variety of answers from a variety of sources.
This Business Insider article, which just references Quora, says

"It depends on how cheap the airline is, but I've had my share seeing them [the cleaning team] refolding the blankets for the passengers on the next flight to use," she wrote. "At least they were generous enough to replace the pillow covers and the headrest covers."

The best solution, according to Busko, is to ask for a new blanket in a plastic bag.

Which makes it sound like reused blankets aren't in plastic bags, if a blanket gets reused (I've never been given a blanket that's not in a bag).

Huffpost seems to corroborate this (although I don't think Huffpost is the best source).

There's also random message boards, and some of these say that the blankets are not reused.

In short: Lufthansa and Condor reuse them and they are washed after every flight. The rest of the answer goes into more detail and gives more sources.

According to a recent article (2017) on travelbook.de Lufthansa reuses them:

Sandra Kraft von der Lufthansa sagt: „Selbstverständlich werden benutzte Decken und Kissen nach jedem Flug eingesammelt und gereinigt.“ Das können Passagiere daran erkennen, dass sie die frischen Decken eingeschweißt an ihren Plätzen finden.

which means (my translation):

According to Sandra Kraft from Lufthansa: "Of course used blankets and pillows are collected and cleaned after every flight." Passengers can recognize this from finding them at their seats packaged.

The article also says similar things about Condor:

Auch bei Condor werden die Decken nach jedem Flug in einer Großwäscherei gereinigt. Die Kissenbezüge dagegen sind Einwegware und werden nach dem Flug entsorgt, sagt ein Firmensprecher.

meaning:

Also at Condor blankets are washed after every flight in a laundry. The pillow cases are disposable however and thrown away after every flight, according to a company spokesperson.

The article goes on to say:

Und wann kommen die Decken und Kissen endgültig weg? Bei Condor werden Kissen in der Economy Class ungefähr 65 Mal frisch bezogen, bevor sie kaputt oder schmutzig sind und deshalb aussortiert werden. Decken verwendet die Airline im Schnitt 24 Mal. Danach werden sie recycelt und anschließend zum Beispiel als Dämmmaterial verwendet.

This says:

And when are blankets and pillows finally thrown away? In Condor's economy class, pillows are newly cased approximately 65 times before they are broken or dirty and thus are disposed of. A blanket is reused approximately 24 times. Then they are recycled and afterwards used as e.g. insulating material.

The article then goes on to say that blankets at Lufthansa are used 30-70 times on average. After that they are donated to charities or also recycled.

The same article appeared on many portals, for example at msn.com and other news portals.

There's also a link to an article from news.com.au from 2016, according to which:

EVER wondered what happens to all those airline blankets and pillows after their flying days are over?

Turns out Qantas has been donating their used blankets and doonas to human services charities for years.

While this is not a direct answer to the question, I thought it's close enough to be included. From the wording of the question though I think one could conclude that also Qantas reuses their blankets.