  We ask if there exists a symmetric chain decomposition of the cuboid $Q_k
\times n$ such that no chain is "taut", i.e. no chain has a subchain of the
form $(a_1,\ldots, a_k,0)\prec \ldots\prec (a_1,\ldots,a_k,n-1)$. In this
paper, we show this is true precisely when $k \ge 5$ and $n\ge 3$. This
question arises naturally when considering products of symmetric chain
decompositions which induce orthogonal chain decompositions --- the existence
of the decompositions provided in this paper unexpectedly resolves the most
difficult case of previous work by the second author on almost orthogonal
symmetric chain decompositions \cite{orth}, making progress on a conjecture of
Shearer and Kleitman. In general, we show that for a finite graded poset $P$,
there exists a canonical bijection between symmetric chain decompositions of $P
\times m$ and $P \times n$ for $m, n\ge \text{rk}(P) + 1$, that preserves the
existence of taut chains. If $P$ has a unique maximal and minimal element, then
we also produce a canonical $(\text{rk}(P) +1)$ to $1$ surjection from
symmetric chain decompositions of $P \times (\text{rk}(P) + 1)$ to symmetric
chain decompositions of $P \times \text{rk}(P)$ which sends decompositions with
taut chains to decompositions with taut chains.
