Is every arrangement reachable by shuffling this way? Suppose we have a vertical stack of $n$ distinguishable coins, each of which is either heads-up or tails-up. Let a shuffle be the following procedure: divide the stack at will into a top- and bottom-stack, and simply rotate the entire top stack $180^\circ$ as a unit. Thus, for $1\le k\le n$: 
$$\underbrace{x_1,...,x_{k}}_{\text{top stack}},\underbrace{x_{k+1},...,x_n}_{\text{bottom stack}} \\
\downarrow\\ \overline{x_k},...,\overline{x_{1}}, x_{k+1},...,{x_{n}}$$
where each $x_i$ is either $H_i$ or $T_i$, and
$$\overline{x_i} = 
\begin{cases} 
H_i,  & \text{if }x_i = T_i \\
T_i, & \text{if }x_i = H_i. 
\end{cases} $$
Since there are $2^n$ conceivable unsubscripted $H/T$ sequences, and there are $n!$ ways to append the subscripts to each, there are $n! \cdot 2^n$ conceivable ways to arrange the stack. 
Is it the case that, for any $n$, all of these conceivable arrangements are reachable by repeated shuffling (no matter what the initial arrangement)? 

Asking whether every arrangement can be reached is the same as asking whether every arrangement can be sorted (into a certain given configuration, such as having the subscripts in ascending order and all coins heads-up).
Other posters have pointed out that this is possible. The question of how to this efficiently is known as the "Burnt Pancake Flipping Problem".  You (a waiter) have a stack of pancakes of all different sizes, but they're out of order--you want to sort the pancakes in order by size, so the biggest one is on the bottom, followed by the next-biggest, and so on.  The only operation available to you is to take a spatula, insert it somewhere into the stack, lift the stack of pancakes above the spatula, invert it, and replace it.
Further, one side of every pancake is burnt, and (for the sake of presentation) you want to put all the burnt sides face-down. (This corresponds to the Heads/Tails in your coin example.)
Various algorithms have been produced to sort a given stack of burnt pancakes, with the goal of using the fewest flips necessary.  Further information:
http://www.theguardian.com/science/blog/2013/nov/14/flipping-pancakes-mathematics-jacob-goodman
http://en.wikipedia.org/wiki/Pancake_sorting#The_burnt_pancake_problem
http://www.math.illinois.edu/~dwest/openp/pancake.html
Why is sorting pancakes NP hard?
http://arxiv.org/pdf/1010.0219v1.pdf
FYI, this problem is related to the study of DNA mutation. Also, you might be interested to know that Bill Gates coauthored a paper on the "unburnt" variation of this problem.
