Symmetry in Pattern Databases
From your question I assume you understand how symmetries are computed. As an exercise, make sure to understand the example given in Figure 4 which refers to the easiest of all symmetries, the Mirror (also known as Diagonal) denoted as $D$. In the following I will refer to this symmetry only unless otherwise noted.

Next, hope you also understand that symmetries create isomorphic puzzles so that the number of moves necessary to optimally solve the reflected state is precisely the same number of moves (but different) necessary to optimally solve the original state ---see Figure 3 in the paper for a good example of a path reflection which results from applying $D$.

Hence, when computing the heuristic estimate of any state $p$, compute its symmetrical state $p'$ (in time linear in the size of the puzzle), and look up the Pattern Database for both states $p$ and $p'$. Since Pattern Databases are indeed homomorphisms let $\rho$ denote such mapping ---i.e., what tiles are mapped to themselves and what tiles are replaced by the don't care symbol. Thus, you can safely take the maximum of both values returned from the PDB:

$h(p)=\max\{V(\rho(p)), V(\rho(p'))\}$

where $V(n)$ denotes the value stored in the PDB for the abstracted state $n$ which consists of tiles and don't care symbols. The two values within brackets are indeed those mentioned at the beginning of Section 4.3.

Note the importance of the blank tile: The other symmetries H, V and D' do not preserve its location and therefore Section 4.2 proposes to use some corrections (3, 3, and 6 respectively) to both the lower and upper bounds.

Of all these symmetries, the most widely used are $D$ and $D'$. Other works (among many others) exploiting them are given below:

Richard E. Korf, Ariel Felner. Disjoint Pattern Database Heuristics. Artificial Intelligence, 134, pp. 9--22, 2002

Ariel Felner, Richard Korf, S. Hannan. Additive Pattern Database Heuristics. Journal of Artificial Intelligence Research, 22, pp. 279--318, 2004.

Uzi Zahavi, Ariel Felner, Robert C. Holte, Jonathan Schaeffer, Duality in Permutation State Spaces and the Dual Search Algorithm, Artificial Intelligence, 172 (4--5), pp. 514--540, 2008

To finish, let me reproduce here a paragraph from the last paper:


  Because all valid, alternative PDB lookups provide lower bounds on the
  distance from state $S$ to $G$, their maximum can be taken as the
  value for $h(S)$. Of course, there is a tradeoff for doing this
  ---each PDB lookup increases the time it takes to compute $h(S)$. Because additional lookups provide diminishing returns in terms of the
  reduction in the number of nodes generated, it is not always best to
  use all possible PDB lookups.


Hope this helps,