Excuse the instrument-geek sidetrack here...
The "octave key" equivalent of the clarinet takes the same fingering from the low register and moves it up a an octave and a 5th (a 12th if you wanna get hardcore about it). The same fingering that gets you low E w/o that key will get you a middle B with it.
So yeah, an F will take you up to a C, but the next/open F# will not get you a C#, at least not one that's in tune. and then up from there, forget about symmetry between registers. And in between the middle G and that middle B, there's G#/Ab, A, A#/Bb, a set of notes known as "the break", and they were the leading bane of my attempts to learn clarinet almost 10 years after playing saxophone exclusively for almost 10 years. Even for people who really play clarinet, "the break" is a very real thing in and of itself. It's a bitch, period.
The objective with any instrument is to have a consistency of timbre across all ranges. On saxophone, it's work, but I could do it. There's an open side-key/closed key fingerings exercise, as well as a throat exercise to change octaves entirely by airflow alone, that Dave Liebman passed on from Joe Allard that is pretty much the go-to exercise to work on that, becuase on saxophone, it's a direct match between octave fingurings, Not "easy" but all it takes is work and ears. On clrintet...there is no such exact symmetry, and to get from a middle Bd (closed keys at all) to a mere half-step up, a B (ALL closed keys) and not have a timbre change....obviously it can be done, but just as obviously to me, I could nevr do it.
Conventional wisdom is to start on clarinet, then learn the saxophone, and conventional wisdom is indeed correct about this, if you want to learn both instruments with any deal of real competence.
I actually played contra-bass clarinet in college symphonic band for a year. Lots of long, low notes, easy as hell to read, but you gotta hit those notes correct and hold them proper, because they're exposed like hell, and if you don't....everybody knows where the fuckup occurred. EVERYBODY. It's not a section! Both it and bass clarinet were easier for me to pick up because the embouchure was a lot closer to tenor, and - the big plus - the had padded keys, not open-holed ones.
Look at this fingering chart, see for yourself what the symmetries are and are not. I get dizzy just looking at it.
Anyway, back to Bennie Maupin, a well-trained musician who no doubt learned it all the right way, and did the right work. Obviously!