"Focal points" of my lenses not aligning?

Hello, and my apologies for what will surely be invented terminology to describe my problem. Thanks in advance for any insight members might provide.

I'm a longtime, successful wearer of multifocal glasses. I have a new prescription from a new optometrist and glasses from a new optician. I'm experiencing what I believe to be a problem regarding what I will call the focal points of the two lenses along the horizontal plane:

1. I look at an object -- e.g. a printed word, at a distance of three feet (happens with other distances as well).
2. I close my right eye, and adjust my gaze so that the word is as perfectly in focus via my left eye as possible.
3. Holding my head still, I close my left eye and open my right eye; the word is now out of focus.
4. In order to bring the word into "optimal" focus for my right eye, I have to rotate my head to the left about four degrees (i.e., effectively changing horizontally the part of the lens I'm focusing through).

Is this to be expected, or is it indicative of some issue that should be corrected? And if the latter, would it be an issue with the prescription or e.g. with the measurements taken by the optician?

Many thanks.