I've skimmed through the article K linked. I found it a fascinating read, but was too impatient to read it thoroughly right now. Anyway, two quotes from it:
In television commercials for laundry detergent, the housewife sensually presses her face into a soft, warm, fragrant, fluffy towel which, along with a tidy bowl, signifies her success as a wife and mother, as a woman, her love for her family. Laundry signifies love."
Yup, we've all seen this image countless times in different variations. This is exactly what Ennis does with Alma Junior's sweater after she left. The way he cares tenderly about a piece of clothing shows his love for its owner. The fact that it's not freshly washed, but still carries her scent, makes it even more poignant, IMO.
"When the ritual [of doing the laundry] is over, our clothes are clean and folded, and a kind of lesser order is restored. Dirt is gone; life starts anew. The bodies we put into these fresh clothes may do something different this time around."
I think this is a very interesting thought. New chances. We haven't discussed this aspect of doing laundry yet. When Jack does the laundry naked after TS1, something new starts. Before they had been friends, but now they are also lovers, a new part of their relationship begins.