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From late January to early February 2005, a word that was perhaps taken from the lyrics to this song appeared on the start page of radiohead.com:
headline
On April 16th 2005, Thom performed an early arrangement of the song at the Trade Justice vigil in London:
'House of Cards' was one of the titles that appeared on the blackboard the band used during two weeks of rehearsals from september 19th to 30th 2005. At the end of that session, Colin posted a picture of this board on Dead Air Space on september 30th, unfortunately (and probably intentionally) in rather low resolution, so that some of it can't be deciphered:
In October 2005, the band worked on 'House Of Cards' during the two-week session. The song appears in two posts from Dead Air Space. The first one shows a lyric sheet that contains the lyrics to 'House Of Cards', in the second Thom mentions listening back to the recording they put to tape during that session:

DAS
DAS
On March 8th 2006, shortly after sessions with Mark Stent had come to an unfruitful end, another photo of a blackboard (and a perhaps slightly worried Ed) was posted on Dead Air Space. A tour had been scheduled for may and june to escape the frustration of the recent sessions and to breathe new life into the project. Rehearsals for the upcoming dates had begun recently, and this board shows which new songs were at this point considered for inclusion in the setlists. 'House of Cards' was one of these, and would indeed be played (click image for full size):
Thom: "There's another song called 'House Of Cards' - that's my personal favourite. It reminds me a lot of that tune by Fleetwood Mac - Albatross, which I absolutely love. House Of Cards sounds satisfying, really mellow and summery..."

Q: "All adjectives you don't normally associate with Radiohead."

Thom: "I know (laughs). We're more associated with terms like 'electronic', 'neurosis', 'madness' and 'fucked-up beats'."
On March 12th 2007, Radiohead's technician Plank mentioned unspecified work on 'House Of Cards' in an entry of his blog at stringsreunited.com:

Wired ?

Monday March 12th

Nice day here today - looks like it's Spring at last ... hooray! Firstly - thanks for all the replies with the L6 diagrams which were useful - what horrible wiring and an over complicated switch - nice guitar though and the switch does add to the appearance once all the wiring has been hidden away...
Today I have mostly been hearing ' House of cards' whilst working up here trying to stop the old Vox combo from rattling.............. a continual problem as all the screws loosen off regularly..... if there is time later next week I hope to make some separate boxes for the amps and speakers.....wishful thinking maybe?..
Just as I was starting to feel settled in, I found out this morning that we will be on the move again at the end of the week to distant location no 473 ! ..... so.. out with the old equipment lists this evening, hopefully catching everybody in the next day or two to fill them in (out?)....
The little HiWatt echo unit is pretty good - not as many features as the old space echo but one neat thing is that it switches itself out when you stop playing, which keeps the background noise down............
pic is of circuit boards from the Emulator - there is less memory on all those boards than on the latest kids mobile phone and the disk drive sticks if you move it anywhere ..ho ho!...... :-)

'House Of Cards' was released as a double A-side radio single with 'Bodysnatchers' in the UK.
Live performances #70 to #74 during the touring for The King of Limbs:
  The King of Limbs tour (1st half) 09/'11 02/'12 march 2012 april 2012 05/'12 june 2012
Bodysnatchers [20] 28 29 27 29 01 05 07 13 15 14 21 29 31 01 03 05 08 10 11 13
Nude [21] 28 29 27 29 01 03 07 11 13 09 12 17 18 21 31 03 05 08 11 13 15
Reckoner [31] 28 29 27 29 01 03 05 06 07 09 11 13 15 09 11 12 14 17 18 21 29 31 01 03 05 06 08 10 11 13 15
House of Cards [5] 18 21 06 08 13
Live performances #75 to #79 during the touring for The King of Limbs:
  The King of Limbs tour (2nd half) july 2012 september 2012 october 2012 november 2012
Bodysnatchers [20] 11 13 15 29 23 29 30 09 11 12 14 15 16 18 06 09 12 13 16 17
Nude [28] 10 11 13 15 25 27 29 20 22 23 26 29 30 06 08 09 11 12 14 15 16 18 06 09 12 13 16 17
Reckoner [29] 10 11 13 15 25 27 29 20 22 23 25 26 29 30 06 08 09 11 12 14 15 16 18 06 09 12 13 16 17
House of Cards [5] 11 25 22 25 16
  2017 A Moon Shaped Pool tour march/april 2017 june 2017 july 2017
Bodysnatchers [22] 30 01 03 08 11 14 17 21 06 09 11 14 16 18 20 23 28 30 02 04 07 19
Nude [17] 30 03 05 09 11 14 18 06 16 18 23 28 30 02 04 07 19
Reckoner [14] 30 05 08 09 17 21 06 16 18 28 30 02 07 19
House of Cards [2] 01 18