This was originally going to be just a top 10, but there are so many that I love and would hate to leave out that I just said "Fuck it, I'm making it a top 20." Anyway, I recommend listening to any you don't know before voting.

Here are the links, tell me if they don't work. I'm sorry that some are either the full version, or don't have the opening video with it, but I'm feeling too lazy and sick right now to search all of YouTube.

Links:
Duvet: link (please ignore the static in the beginning, I couldn't find a video without it)

Guren No Yumiya: link

Monochrome no Kiss: link

Connect: link

Kuusou Mesorogiwi: link

Naraku No Hana: link

Calendula Requiem: link

Higurashi no Naku Koro ni: link

Full Colors: link

Kyomu Densen: link

My Soul, Your Beats!: link

The World: link

Tori no Uta: link

Unravel: link

Uragiri no Yuuyake: link

Lilium: link (I am so so sorry about the mature picture in this one!)

Sakasama no Chou: link

Hello Sleepwalkers: link

In my World: link

Guardian Angel: link

I also liked Guren No Yumiya, The World, Colors and My Soul, Your Beats. But In My World is my 2nd Anime Opening.

Wait, so the links work? Just checking, since last time I used links I failed miserably . . .

Also my favorite opening theme. <3
Judging by your list we also seem to have similar taste in anime music. ^^

Yet again, I'm not surprised!
I honestly used to slightly dislike this theme, but after rewatching the series a few times it really grew on me.

Loved this opening ever since October 14, 2013.

I also love Calendula Requiem, Kyomu Densen, Guren no Yumiya, Kuusou Mesorogiwi, Hello Sleepwalkers, Uragiri no Yuuyake, Higurashi no Naku Koro ni, and My Soul Your Beats!.

Love your taste in openings,my top five is in your top twenty I think besides one! Love this animation.