And as the system is very new, and doesn't have personal files on it yet, I can easily create an account if someone needs to debug remotely...

I suspect there's in issue with the xrandr 1.1 compatibility stuff in the xserver since you are using xrandr -o and your laptop has several outputs.  Do you still get the crash if you specify the rotation with xrandr 1.2 API calls?

xrandr --output LVDS --rotate left

Still dies with a sequence of 

xrandr --output LVDS --rotate left
xrandr --output LVDS --rotate right
xrandr --output LVDS --rotate normal
xrandr --output LVDS --rotate inverted

After some number of these, the X server is still crashing...

Can you start the xserver in gdb rather than attaching to the running process?  It'd be nice to get a backtrace.  Unfortunately, I can't reproduce it here.  It looks like it may be related the dixLookupPrivate changes.

OK, that worked.  Here's the backtrace...

#0  dixLookupPrivate (privates=0x198, key=0x81df2d8)
    at ../../dix/privates.c:131
	rec = <value optimized out>
	ptr = <value optimized out>
#1  0x080c871b in xf86ResizeOffscreenLinear (resize=0x96a7c58, size=1024000)
    at ../../../../hw/xfree86/common/xf86fbman.c:199
No locals.
#2  0xb79f0f77 in radeon_allocate_memory (pScrn=0x968bf58, 
    mem_struct=0x96a6480, size=4096000, align=4096)
    at ../../src/radeon_memory.c:61
	linear = <value optimized out>
	pScreen = (ScreenPtr) 0x96a8458
	info = <value optimized out>
	offset = <value optimized out>
#3  0xb7a1615e in radeon_crtc_shadow_allocate (crtc=0x96a62a0, width=1280, 
    height=800) at ../../src/radeon_crtc.c:421
	pScrn = (ScrnInfoPtr) 0x198
	rotate_offset = <value optimized out>
#4  0x080f40c8 in xf86CrtcRotate (crtc=0x96a62a0, mode=0xbfa89870, rotation=4)
    at ../../../../hw/xfree86/modes/xf86Rotate.c:607
	shadowData = <value optimized out>
	shadow = (PixmapPtr) 0x0
	width = 1280
	height = 800
	old_width = 0
	old_height = 0
	pScrn = <value optimized out>
	pScreen = (ScreenPtr) 0x96a8458
	crtc_to_fb = {matrix = {{-65536, 0, 83886080}, {0, -65536, 52428800}, {
      0, 0, 65536}}}
	fb_to_crtc = {matrix = {{-65536, 0, 83886080}, {0, -65536, 52428800}, {
      0, 0, 65536}}}
#5  0x080eb243 in xf86CrtcSetMode (crtc=0x96a62a0, mode=0xbfa89870, 
    rotation=408, x=0, y=0) at ../../../../hw/xfree86/modes/xf86Crtc.c:312
	scrn = (ScrnInfoPtr) 0x968bf58
	i = <value optimized out>
	ret = <value optimized out>
	didLock = 0
	adjusted_mode = (DisplayModePtr) 0x9897d40
	saved_mode = {prev = 0x0, next = 0x0, name = 0x0, status = MODE_OK, 
  type = 0, Clock = 71450, HDisplay = 1280, HSyncStart = 1301, 
  HSyncEnd = 1333, HTotal = 1460, HSkew = 0, VDisplay = 800, VSyncStart = 804, 
  VSyncEnd = 808, VTotal = 816, VScan = 0, Flags = 10, ClockIndex = 0, 
  SynthClock = 0, CrtcHDisplay = 1280, CrtcHBlankStart = 1280, 
  CrtcHSyncStart = 1301, CrtcHSyncEnd = 1333, CrtcHBlankEnd = 1460, 
  CrtcHTotal = 1460, CrtcHSkew = 0, CrtcVDisplay = 800, CrtcVBlankStart = 800,
   CrtcVSyncStart = 804, CrtcVSyncEnd = 808, CrtcVBlankEnd = 816, 
  CrtcVTotal = 816, CrtcHAdjusted = 0, CrtcVAdjusted = 0, PrivSize = 0, 
  Private = 0x0, PrivFlags = 0, HSync = 0, VRefresh = 0}
	saved_x = 0
	saved_y = 0
	saved_rotation = 1
#6  0x080f2582 in xf86RandR12CrtcSet (pScreen=0x96a8458, randr_crtc=0x96a90a8, 
    randr_mode=0x96a9930, x=0, y=0, rotation=408, num_randr_outputs=1, 
    randr_outputs=0xbfa89a30) at ../../../../hw/xfree86/modes/xf86RandR12.c:769
	mode = {prev = 0x0, next = 0x0, name = 0x0, status = MODE_OK, 
  type = 0, Clock = 71450, HDisplay = 1280, HSyncStart = 1301, 
  HSyncEnd = 1333, HTotal = 1460, HSkew = 0, VDisplay = 800, VSyncStart = 804, 
  VSyncEnd = 808, VTotal = 816, VScan = 0, Flags = 10, ClockIndex = 0, 
  SynthClock = 0, CrtcHDisplay = 1280, CrtcHBlankStart = 1280, 
  CrtcHSyncStart = 1301, CrtcHSyncEnd = 1333, CrtcHBlankEnd = 1460, 
  CrtcHTotal = 1460, CrtcHSkew = 0, CrtcVDisplay = 800, CrtcVBlankStart = 800, 
  CrtcVSyncStart = 804, CrtcVSyncEnd = 808, CrtcVBlankEnd = 816, 
  CrtcVTotal = 816, CrtcHAdjusted = 0, CrtcVAdjusted = 0, PrivSize = 0, 
  Private = 0x0, PrivFlags = 0, HSync = 0, VRefresh = 0}
	pScrn = (ScrnInfoPtr) 0x968bf58
	crtc = (xf86CrtcPtr) 0x96a62a0
	changed = 1
	o = <value optimized out>
	ro = <value optimized out>
	save_enabled = 1
#7  0x0816ce96 in RRCrtcSet (crtc=0x96a90a8, mode=0x96a9930, x=0, y=0, 
    rotation=<value optimized out>, numOutputs=1, outputs=0xbfa89a30)
    at ../../randr/rrcrtc.c:302
	pScreen = (ScreenPtr) 0x96a8458
	ret = <value optimized out>
#8  0x08171f1b in ProcRRSetScreenConfig (client=0x9899108)
    at ../../randr/rrscreen.c:922
	rep = {type = 8 '\b', status = 145 '\221', sequenceNumber = 2441, 
  length = 136311872, newTimestamp = 3215497816, 
  newConfigTimestamp = 135622986, root = 136311876, subpixelOrder = 39476, 
  pad4 = 49064, pad5 = 159968648, pad6 = 23}
	pDraw = (DrawablePtr) 0x96fec20
	rc = <value optimized out>
	pScreen = (ScreenPtr) 0x96a8458
	pScrPriv = (rrScrPrivPtr) 0x96a9f78
	time = {months = 0, milliseconds = 8555932}
	i = 0
	rate = 60
	has_rate = 1
	output = (RROutputPtr) 0x96aa128
	crtc = (RRCrtcPtr) 0x96a90a8
	mode = (RRModePtr) 0x96a9930
	pData = (RR10DataPtr) 0x96e72f8
	pSize = (RRScreenSizePtr) 0x96e730c
	width = 1280
	height = 800
#9  0x0816b5a5 in ProcRRDispatch (client=0x198) at ../../randr/randr.c:476
No locals.
#10 0x0808c91f in Dispatch () at ../../dix/dispatch.c:454
	result = <value optimized out>
	client = (ClientPtr) 0x9899108
	nready = 0
	start_tick = 3400
#11 0x08071d8d in main (argc=1, argv=0xbfa89ba4, envp=Cannot access memory at address 0x1a0
) at ../../dix/main.c:441
	i = <value optimized out>
	error = 136146932
	xauthfile = <value optimized out>
	alwaysCheckForInput = {0, 1}
(gdb) 

FWIW: this time, before crashing, it put some trash on the screen; clearly pixmaps that don't belong on the screen....

Something is trashing memory allocation.

Does this also happen with EXA rather than XAA?

Yes.  Still crashes.

if you are running a git based version of ati can you try again with 6.9.0?

Is this still an issue with xf86-video-ati from git master or a newer xserver?

closing due to lack of feedback.