There is a dispute about whether strength or writing is the deciding factor in determining which hand will be used to wear tefillin.
Therefore, some who perform all labor with one hand and write with the other, have the practice toward the end of davening to remove the shel yad and put it on the opposite arm for a few minutes to be mekayem both shitos.
Just daven normally, if people can talk on the phone to their friends in public and wear all kinds of crazy clothes, we can talk to the Creator in public wearing tefillin etc.
Blessings and success
Trying to turn a discussion in comments into an answer:
The Shulchan Aruch and the Rema (27(1)) poskin like the opinion that
the arm tefillah should be put on the half of the muscle toward the
elbow. (The Gr"a points out that every other Rishon argues and
allows the entire muscle. And the Aruch Hashulchan there actually
holds that the entire upper arm ...
The poskim - see for instance Rambam Hilchos Tefillin 2(11) - bring a source (Yerushalmi last perek of Eiruvin - Kesef Mishnah) that Hillel (some say Shammai) used his grandfather's tefillin without needing to check them. That is part of a discussion there whether tefillin need checking in general, or how often. I don't know that it had anything to do with ...