I do a lot of GIS work with Alaska data, much of which comes from the State Geo-data Clearinghouse in EPSG: 3338 (NAD83 Alaska Albers), which has a longitude of false origin of -154. Because Alaska is such a large area, making maps away from that longitude results in a skewed appearance where N, S, E, W lines are angled. Here's an example from the northwest part of the state, using the default longitude value (-154) where the black lat/long lines are 68, -163:
In order to make such maps more pleasing to my audience, I often modify the CRS' longitude of false origin to match the longitude of the map center. This can easily be done in either the QGIS Project Properties or ArcMap Data Frame Properties. In the following example, I've changed the longitude of origin from the default -154 to match the map's -163:
I store the data, and do all analysis with the default CRS. Modifying the latitude of origin is only done for final map production and display.
Obviously, the "non-square" effect occurs in other locations and with other CRSs. I use Alaska for illustrative purposes.
AFAIK, there is no drawback to adjusting the longitude of origin for display. To me, what I'm doing is akin to rotating a globe while I look at it, holding my eyes steady. But still, is there something here that is cartographically improper?