Please understand that you've asked the designer of one loudspeaker to critique it against another. I like the other design, and actually, I like it quite a bit. But my opinion is naturally biased heavily towards my own design. Having stated that qualification, I'll continue to tell you my thoughts.I really love the sound of a nice pair of Klipschorns. The late seventies models of Klipschorns are really their best, in my opinion, and a pair in good shape are the best representative of them. They are a very satisfying speaker, when installed in appropriate corners.
On the other hand, there are some model years of Klipschorns that don't sound good at all. Some have too much midrange and others have muddy bass. Their cabinets are complex, and if they have been pushed hard or mistreated at all, they can rattle and buzz. So comparing Klipschorns to π cornerhorns really requires us to define models and conditions.
Similarly, π cornerhorns have been available with several drivers over the years. Just about everything from $35.00 10" woofers to $800.00 18" woofers have been used in this configuration. So, again, one really has to qualify this part for the best discussion.
I think π cornerhorns with high quality 15" and 18" pro drivers are substantially better than the best Klipschorns. It isn't really a fair comparison though, because the Klipschorn design is made to use drivers that aren't terribly expensive. Some Klipsch fans cry that this is because the design limits excursion and increases efficiency so a better driver isn't required for best system performance. I think there is some truth to this, but I also think it is somewhat of a justification and that systems using woofers with better motor structures outperform it.
But generally, when comparing the two designs having components of similar quality and cost, I think they are very similar in midbass character, provided they are both installed in good, solid corners. At the lowest frequencies, I think the Klipschorn sounds full, but can also sound a bit resonant and boomy. The π cornerhorn sounds more natural, and this is particularly noticable in bass note progressions, where response is flat, without one note being more prominant than others.
At the upper extreme, into the lower midrange frequencies, the π cornerhorn sounds very true and free of box or quarter-wave chamber resonances. There is no throat distortion and no labyrinth for the lower midrange to pass through. The wavelengths are a few feet long at lower midrange frequencies, so a π cornerhorn acts as a simple conical horn with the diaphragm very close to the apex. But the Klipschorn has several passages that become near 1/4 wavelength in this region, so it starts sounding a bit throaty. I think that's why one of the "holy grails" for the Klipsch guys is to find midrange compression horns with a low frequency cutoff.
A Klipschorn is very efficient, but it is also somewhat peaky and highly dependent on the room's corner to achieve flat response. If the corners aren't right, then the labyrinth internal to the Klipschorn cabinet becomes a severely truncated horn, and response becomes noticeably peaky. So the π cornerhorn is more forgiving of its environment, and if the corners aren't quite right, they might lack bass a little bit but they will not become resonant or boomy in the midbass.
Both designs are very good, in my opinion. Both configurations depend on room boundaries, and really, the room's corner provides most of the horn flare in either case. But I think the π cornerhorn is capable of cleaner output, and particularly when using more expensive components, is a much higher fidelity speaker and more powerful too.