Karakuri Benihime Kaikuguri[絡繰紅姫攫い潜り]
Karakuri[絡繰]: Mechanical Doll
Benihime[紅姫]: CrimsonPrincess
Kai[攫い]: Abduction/Kidnapping
kuguri[潜り]: Side door/gate
Karakuri isn't a general term, btw. It's a specific kind of traditional Japanese puppet that became popular in Japan in the 1600s. Some were utilized as curiosities that could preform singular and at times complex tasks like serving tea, or writing simple calligraphy, or shooting a bow and arrow at a target. But more relevantly they were used in theater and in parades to act out mythic or other dramatic scenes.
More colloquially it can also refer to a general string puppetry as well, which seems to be what the design here is more referencing. This and the actual bankai both draw on Japanese balljointdolls as an aesthetic, but it should be noted that the more important aspect tying these two types of puppets together is the aspect of control, and not the doll quality itself.
The kuguri bit is also a specific reference to a kind of Japanese architecture, and not just anything fitting the the literal description. The actual etymological root is the character for "concealed"/"hidden"/"quiet" rather than "side." It's probably meant as a play on Urahara's tendency throughout the series to backdoor his way into places he's not supposed to be.
Sakae Madowashi Senkohou[逆廻惑閃光砲]
Sakae[逆廻]: Reverse + Rotation
Madowashi[惑閃]: to bewilder/perplex/puzzle/delude/ mislead/deceive (but also derived from Klesha*) + Flash
Senkohou[光砲]: Light + Cannon
Not that it's been uncommon for Bankai to change names, but it seems odd that this is now the third time from Sakanade[逆撫]: "Reverse+Pet(Irritate)", to Sakashima[逆様]: "Reverse Order," now to Sakae[逆廻]: "Reverse Rotation."
Anyway... this one's a little more straight forward...
*I was gonna try and summarize this myself but the wiki page covers it concisely enough:
Kleshas, in Buddhism, are mental states that cloud the mind and manifest in unwholesome actions. Kleshas include states of mind such as anxiety, fear, anger, jealousy, desire, depression, etc. Contemporary translators use a variety of English words to translate the term kleshas, such as: afflictions, defilements, destructive emotions, disturbing emotions, negative emotions, mind poisons, neurosis etc.
So, a flash of emotional or mental or spiritual disruption inflicted by a cannon blast of literal light. (Not as if it isn't commonly used in other kidou, but it's the same [閃]:"flash" Kubo uses for Cero[虚閃] which feels like it might be relevant to Shinji having his mask on for this.)
On the one hand the more literal take on this is that it's a blast of light that confuses the opponent, but given the rotational motif and directional reversal, in conjunction with the buddhist themes this seems to imply reversing the progress one makes toward enlightenment; a kind of regression of development. And technically that's kind of what the whole hollowization process is, a conflict with the inner self, the opposite of enlightenment.
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