Recursively Rotated Possible Permutation

[Leroy Quet]

I just submitted the following sequence to the EIS.

%S A000001 1,3,4,2,7,8,6,11,12
%N A000001 Start with positive integers. On m_th step shift terms a(m+1) 
to a(2m-1) one position to the left, move a(m) to position (2m-1). 
Sequence is limit of reordering.
%e A000001 [1,2,3,4,5,6,...]->[1,3,2,4,5,6,...]->[1,3,4,5,2,6,...]
->[1,3,4,2,6,7,...]->[1,3,4,2,7,5,...]->[1,3,4,2,7,8,...]
->...
%O A000001 1
%K A000001 ,more,nonn,

Is it a permutation of the positive integers?
(Or do some integers get rotated out to infinity?)

I bet there is a simple, even trivial, way to express which integer is at 
any particular position. (But a maybe not.)

Could someone please extend this sequence once it appears, and if it is a 
permutation, submit the inverse?

thanks,
Leroy Quet