In the 5th Commandment, we are ordered to respect our father and mother. However, in Leviticus 19:3, we find the order with a twist… mother first and then father. Is there any meaning to the swiching of order?

That is a very astute observation. Indeed, the Talmud (Kidushin 30b-31a) does deal with your question. There, the great sage Rabbi Yehuda HaNassi explains that, generally, a mother shows more love and kindness to her child. Since it is, therefore, more natural for a child to show honor to the mother, the Torah emphasizes the need to honor the father as much as the mother. On the other hand, Leviticus 19:3 deals with fear, or awe, of parents. In that context, the Torah places the mother first, since a child is more accustomed to being around her and is more naturally in awe of the father. Together these verses teach us that the father and mother are equal both in terms of honor and in terms of awe (Maimonides, Hilchos Mamrim, 6:2).

Sincerely,
R’ Daniel Fleksher