First, I consulted my letter distribution for Bananagrams to see which letters occurred an odd number of times in a Banangrams set. For instance, there are three W tiles, so I knew that at least one (and possibly all three) would have to occur on the center line of the grid to maintain mirror symmetry.
I then tried to build those letters into vertical words. Next I built some horizontal palindromes off of those center-line words. I got some good palindromic words from Wiktionary.
Finally, I worked on linking everything up. This step took the longest time. Fortunately, I found a list of words, that when reversed, yield different English words (like "tuba" and "abut"). These are called "semordnilaps". I particularly liked the ERGO/OGRE pair, so I managed to work that one into my grid. After a lot of fiddling around, I wound up with the grid shown below.