Solution:

Imagine that the rooms are colored black and white, checkerboard-style. Each guest in a black room moves to an adjacent white room (and vice versa). If, for each such guest, we place a domino over the original room and the new room, we obtain a covering of the $2 \times n$ grid by $n$ dominoes, since each black square is used once and each white square is used once. Applying a similar procedure to each guest who begins in a white room and moves to a black room, we obtain a second domino tiling. Conversely, it is readily verified that any pair of such tilings uniquely determines a movement pattern. Also, it is easy to prove by induction that the number of domino tilings of a $2 \times n$ grid is the $(n+1)$th Fibonacci number (this holds for the base cases $n=1,2$, and for a $2 \times n$ rectangle, the two rightmost squares either belong to one vertical domino, leaving a $2 \times (n-1)$ rectangle to be covered arbitrarily, or to two horizontal dominoes which also occupy the adjoining squares, leaving a $2 \times (n-2)$ rectangle to be covered freely; hence, the numbers of tilings satisfy the Fibonacci recurrence). So the number of domino tilings of a $2 \times 8$ grid is $34$, and the number of pairs of such tilings is $34^2 = 1156$, the answer.