cout << i++ << i--
is semantically equivalent to
operator<<(operator<<(cout, i++),   i--);
           <------arg1--------->, <-arg2->
$1.9/15- "When calling a function
  (whether or not the function is
  inline), every value computation and
  side effect associated with any
  argument expression, or with the
  postfix expression designating the
  called function, is sequenced before
  execution of every expression or
  statement in the body of the called
  function. [ Note: Value computations
  and side effects associated with
  different argument expressions are
  unsequenced. —end note ]
C++0x:
This means that the evaluation of the arguments arg1/arg2 are unsequenced (neither of them is sequenced before the other).
The same section in the draft Standard also states, 
If a side effect on a scalar object is
  unsequenced relative to either another
  side effect on the same scalar object
  or a value computation using the value
  of the same scalar object, the
  behavior is undefined.
Now there is a sequence point at the semicolon at the end of the full expression below
operator<<(operator<<(cout, i++), i--);
                                      ^ the interesting sequence point is right here
As is clear, evaluation of both arg1 and arg2 lead to side effect on the scalar variable 'i', and as we saw above, the side effects are unsequenced.
Therefore the code has undefined behavior. So what does that mean?
Here is how 'undefined behavior' is defined :) in the Standard.
Permissible undefined behavior ranges
  from ignoring the situation completely
  with unpredictable results, to
  behaving during translation or program
  execution in a documented manner
  characteristic of the environment
  (with or without the issuance of a
  diagnostic message), to terminating a
  translation or execution (with the
  issuance of a diagnostic message).
  Many erroneous program constructs do
  not engender undefined behavior; they
  are required to be diagnosed.
Do you see correlation with @DarkDust's response 'The compiler is even allowed to set your computer on fire :-)'
So any output you get from such a code is really in the dreaded realm of undefined behavior.
Don't do it.
Only thing that is defined about such code is that it helps OP and many of us get lots of votes (if answered correctly) :)