What to add to clay. Anything organic. Some people would add granite dust or chips or gypsum, but I just add organic: compost, etc.

Actually, my favorite is soil conditioner. It's sometimes called "Nature's Helper." It is ground up pine bark... or something like that. I use it a lot.

Here are some basics about amending clay (and keep in mind that daylilies do well in plain old clay):
Clay is rich in nutrients (and red clay is rich in iron), so the purpose is to improve drainage. Your other option is to stick with plants that don't need good drainage, but that's a short list.
There is controversy about whether sand improves the drainage or just makes the clay worse. I don't know the answer to that one. I have used sand on occasion, but when I use it, I also work in lots of rotted pine bark and compost and I only use it in small spots. The only time I used sand extensively, I mixed it half and half with soil conditioner and tossed in a bag of gravel (I was really serioius about the drainage in that spot). Daylilies are not going to need this kind of drainage. Nothing I grow except mediterranean plants and daphne need this kind of drainage.
For most new beds, I put an inch or two of peat moss, around four inches of soil conditioner and an inch of cow manure on top of the soil and mix it in. This usually gets the bed off to a good enough start. This will all turn back into clay sooner than you think, so once a year - usually in the fall - I mulch the beds with compost or rotted pine bark and cover it with pine straw. Over the years this loosens up the soil more and more. If I am running out of mulch or time or energy, the one bed I skip is the daylily bed because I know they will do fine in clay.
KathyGA uses an approach I am going to try. She amends only the planting holes, the puts down newspaper and mulch over the rest of the bed and lets nature amend the rest of the soil. Maybe she'll chime in and say what she uses for the mulch.
I am gardening more and more in lasagna beds, but I wouldn't waste a lasagna bed on daylilies. I use them for plants that are less tolerant of clay. So far, everything I plant in lasagna beds thrives. I am a fan of them.

The definition of "fan" is in the section about parts of a daylily (under "leaves").

Here is a link that might be useful: Daylily fact

There really isn't a controversy about the sand and clay issue. Unless you practically replace the whole soil volume with sand, you'll make matter a whole lot worse! Organic amendments will do you just fine. I grow in hard hard red clay, and after just one organic amendment incorporation, I've been HUGELY pleased with how plants grow in clay. Just keep an organic mulch on top, and most plants will do just fine.

Remember, there is a difference between poorly drained soil and clay soil. Clayey soils do not have to be considered poorly drained. My yard drains well in the back (thank goodness), but we had to add some drainage pipes in the front.

I did double dig some of the flower beds in my yard. I had new construction and they left the yard a mess, no top soil. Basically what I did is break up the top 6 inches, got rid of any rocks or that grey clay that may as well be a rock. I mixed what was left with natures miracle. You also have to go down an additional 6-12 inches after removing the top 6 inches. A spade fork works well for this, and it's best to do this when it has rained and the soil is not rock hard.

After that I would put the ammended soil back and put some bags of gardensoil on top. The main thing is to make sure you aren't creating a big clay pot (hole) and filling it up with good dirt. If you do that you will get what is known as the 'swimming pool' effect. the water will stay in the hole and not drain. Then your plants will be sitting in water.

If you are planting something like daylillies, you could simply place them down and cover with top or gardensoil and add lots of mulch. After 2 years they will break up the soil for you.

I would think the fan is the base of the leaves where they 'fan' out.

Oh ESH_GA, that daylilly link you posted above is great, thanks for the great information!