So I have a rationale for why we start with them in the left hand...
Shulchan Aruch, Orach Chaim 24:4 writes:
מצוה לאחוז הציצית ביד שמאלית כנגד לבו בשעת קריאת שמע רמז לדבר והיו הדברים האלה וגו' על לבבך
It is a מצוה to grasp the ציצית with one's left hand near one's heart while reciting Kriyat Shema. This is alluded to [in the verse] "And these words shall be...on your heart."
(Sefaria translation)
(Parenthetically, the Mishnah Berurah at this point tells us about taking the tzitzis also in the right hand during the third paragraph of Shema:
ביד שמאלית - כתוב בכתבי האר"י ז"ל שיאחזם בין קמיצה לזרת וכשיגיע לפרשת ציצית יקחם גם ביד ימין ויביט בהם ויהיו בידו עד שמגיע לנאמנים ונחמדים לעד ואז ינשק הציצית ויסירם מידו
With your left hand - It is written in the writings of the Arizal that [one should] hold them (the Tzitzit strings) between the ring and pinky fingers. And when you get to the Parsha of ציצית, take them also in your right hand and gaze at them. And they should remain in your hands until you reach [the words] "נאמנים ונחמדים לעד"... at which point you kiss the Tzitzit and release them from your hands.
(Sefaria translation))
So we see that part of the rationale for at least starting holding the tzitzis in the left hand is because it is opposite the heart which hints how we should make the words "על לבבך" "on your heart". Also refer to the Shulchan Aruch HaRav 24:4
This point is developed even further by Eishel Avrohom 24:1 which adds:
לכן יטול ביד שמאל נגד לבו וכתיב בהו ולא תתורו אחרי לבבכם כי ציצית מצילין מחטא זנות
Therefore he takes them (the tzitzis) in the left hand opposite one's heart as it writes about it "So that you do not follow after your heart" (Bamidbar 15:39) because tzitzis saves a person from the sin of immorality.