I davvened in a shul in England and spat during the prayer of Aleinu, so the president of the shul came and shouted at me, and asked me what kind of custom is that? It disgraces the shul! He would like to know the source of the custom.
There are no hard and fast rules about white shirts. In general, a white shirt is considered to be a modest but respectable garment, and a person should wear clothing that is respectable.
Also, part of the day is spent standing in prayer in front of the Creator, and this certainly warrants respectable attire.
In any case, the main thing is not the external garments, but rather the spiritual ones, namely, a person’s thought, speech, and deed. Physical clothing is only a means to the end, the goal being purity of spiritual garments.
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