In the Code of Jewish Law1 it is written that a Torah scroll should be held in one's right arm (and rested on the right shoulder). This applies even if the one holding the Torah is left-handed. There are two verses that allude to this idea:

1) "From His right hand was a fiery Law for them" -- Deuteronomy 33:2. The Torah was given from G‑d's right hand, as it were, and we therefore hold it in our right hand, too.

2) "And His right hand will embrace me" -- Song of Songs 2:6. G‑d embraces us with His right hand, and we, in turn, embrace His holy Torah with our right hand.

According to Kabbalah, G‑d's "right hand" is an anthropomorphic metaphor for G‑d's attribute of Kindness, as opposed to the left hand which alludes to the divine attribute of Severity. The Torah was given to us with lovingkindness, and with it G‑d lovingly embraces us. We reciprocate by approaching the Torah with a deep love and appreciation, which even expresses itself in the way we carry it around!