Recommend me some weird music

Yakuza are pretty weird.

The band Puya aren't jarring noise-makers or anything, but the combination of hip-hop, salsa, jazz, samba, hard rock, punk, and metal(w/ solos), all with a twist(español) is something weird and exciting.
 
Another weird album is the german album of Timo Rautiainen ja Trio Niskalaukaus. He´s got a funny pronounciation and it´s hard to understand
what he´s talking about as they translated line for line from finnish into german.
But still the music is great!!!
 
Swans - Excellent experimental music. I highly recommend God Damn The Sun, Let It Come Down, and God Loves America.

Other bands to check out... Tenhi, The Legendary Pink Dots, Úlfur, and In Slaughter Natives.
 
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Many of the following artists/albums may not be regarded as (really) weird, but I think they are at least "original":
Kate Bush - "The Dreaming" (Definitely her weirdest album - and my favourite one! I also like the second part of her "Hounds Of Love"-album very much.)
The Bulgarian Voices "Angelite" feat. Huun-Huur-Tu & Sergey Starostin - "Fly, Fly My Sadness" (I think "Legend" is definitely the best song on this album.)
Atrox - "Contentum" (weird vocals), "Terrestrials" (weird vocals and music), "Orgasm" (not as weird as its predecessor, but still very original)
The Third And The Mortal
Anacrusis
Sainkho - "Naked Spirit"
Hulu Project feat. Stepanida - "TranceSiberia"
Mekong Delta - "The Music Of Erich Zann"
Sieges Even - "Steps"
Hedningarna - "Trä", "Hippjokk"
Tactile Gemma
Watchtower - "Control And Resistance"
Voivod - "Killing Technology", "Dimension Hatröss", "Nothingface"
Garmarna - "Guds Spelemän", "Vedergällningen"
Terje Isungset - "Iceman is" (music played by instruments made of ice), "Reise", "Floating Rhythms"
Fear Of God - "Within The Veil" (definitely one of the most original metal albums ever!)
Psychotic Waltz - "A Social Grace", "Into The Everflow"
Egschiglen - "Zazal"
Mari Boine - "Eallin"
Yat-Kha
Primus
 
I'm not going to get into the specific albums and such, but here are a few very original artists that I think are worth checking out:

Lacuna Coil
Nevermore
System of a Down
Radiohead
Mazzy Star
Mineral
Smashing Pumpkins
Nine Inch Nails

*points to above list* See!? Proof that I do sometimes listen to something other than metal. Sometimes.
 
Estradasphere - You can't get any weirder than these guys, Candiria pales in comparison. They mix death metal, jazz, big band, oldies, hip-hop, funk, NY hardcore; you name it. This band is a must if you want weird music.
 
Japanese Noise bands
- Acid Mother's Temple
- 00I00 (it's the band set up by Yoshimi, most famous in western society for her contibution to the Flaming Lips newest album)

Some of the slowest, most painful music you'll ever hear
- Khanate

No Wave Girl band
-Erase Errata.
 
Deus Ex Machina (from Italy). The lyrics are sung
in Latin. Pretty cool, actually.
 
Seriously, you all need to check out Scott Walker's Tilt. It's the most emotionally wrenching (which is odd since it's very hard to emphasize with anything on the album) and desolate album I have ever heard.

Also some stuff I haven’t mentioned yet:
Stormy Six- L’Apprendista
Univers Zero (get Heresie now!!!)
Doctor Nerve- Skin
Iran- s/t
Iannis Xenakis- Kraanerg
Autechre- Confield
Lightning Bolt- Wonderful Rainbow
Peter Brotzmann- Machine Gun
Naked City- Black Box