Does a left-handed man have to start wearing his clothes from the left side - because his important hand is left, or from the right side - because this is an important hand for most of people?
Like everything else that is done for Kabbalistic reasons, a lefty does the same as a righty.
Which hand a left-handed person should use, depends on which reason a right-handed person is supposed to use their right hand: either because their right hand is more prominent, or because Kabbalistically, the right hand represents the force of Chessed.
The Mishna Berura (4:22) writes that when washing one’s hands, one washes one’s right hand first irrespective as to whether one is right or left-handed. Likewise, everyone should put their right sleeve, etc. in first when getting dressed (See Mishna Berura 2:4). Unlike right-handed people who tie their left shoes first, however, left-handed people should tie their right shoes first (corresponding to which arm they would tie their Tefillin on).