I have a bale of either hay or straw that was used for fall decorations. I was going to put that in. It's been sitting all winter, wouldn't any seeds have already gone?
Faith--most likely what you have there is straw, which is commonly used for decorations. There won't be many seeds in it, and the seedheads that are there are usually wheat seeds. If these sprout in your compost it is easy to turn it over and re-compost the sprouts back in. Some could survive the composting process and sprout in your garden, but that's not really a problem--wheat isn't a virulent pest, lol!
My husband grew up in Philadelphia and he always thought hay and straw were the same thing; now he still gets them confused. BIG difference between them, though!
STRAW is basically a waste product, so it's usually fairly cheap. It is the dead stalks of a grain crop, cut after the grain heads dried and were harvested. Most likely it doesn't have seeds in it except for some from the grain (usually wheat). For animals, straw is used for bedding as it is absorbant. Straw is a great mulch as it is very insulating. In drier climates, straw bales can be used in building construction (with stucco over them). Nutritionally, it is mostly just cellulose. Dead, dry, carbon, like leaves. In the diet of your compost bin, straw can contribute to the large amount of "brown" matter you need as food for your microbes.
HAY is basically a food product, so it's usually more expensive than straw. It is the whole plant (of alfalfa, grasses, and likely a mix of these) which was cut live at its peak and left to dry before bundling. It's dehydrated food just like the zucchini or basil you might dehydrate for storage. If you look at a bale of hay, it often has a greenish tinge, compared to the yellow/brown color of straw. That's because hay IS green, even though dry to the touch, it's a nitrogen source. It's just like your grass clippings from your lawn--you can add it to your compost and it will contribute nutrients as "green" matter but needs a lot of "brown" matter to cook. For animals, hay is nutritious feed and for that reason after a hard winter with a lot of snow cover, farmers will have bought up a lot of the hay and the price goes up. You wouldn't use hay as animal bedding, obviously, as it will get stinky when wet, just like your kitchen scraps or grass clippings do.
Re: weed seeds
Wheat is an annual crop, usually planted in a plowed field, possibly managed with herbicides although often with wheat this is not necessary, as it is planted so densely, and fall planting is common which means the crop is well established before the weeds can sprout. Because of that, straw comes from a relatively 'clean' field with few vicious weeds in it.
Hay, on the other hand, is grown year after year on the same, unplowed ground, as a perennial crop. Like a pasture. Because of that, it is easy for pernicious weeds to get established. So your bale of hay might have many different weed seeds in it. A good hay farmer will be very careful about watching for weeds, especially any which might be harmful for livestock to eat. But it is still an issue for people.
The hay I use for mulch is pretty good as far as weeds go; I've been buying from the same farmer for years. Also on my garden beds I keep adding more layers constantly so weeds that do sprout just die again under the thick mulch. I never have any bare ground so they don't get a chance. The reason I choose hay over straw is A) its nutritional content is great for my soil and B) straw tends to blow away in the high winds we get here. The alfalfa in the hay is 'clingy' and sticks to itself and seldom blows.
Ruth Stout used hay but she also wrote over and over in her books that almost anything will work as mulch--the idea is to protect the soil and let the worms farm it for you!