Music wise, The Redowaczka is basically a Galopade and a music change to 2/4 time. When the polka became the vogue, the Redowa and polka merged to become the Redowa-Polka or Half and Half. The Spanish had their "Half and Half's as well. The Menuet Afandangado is partly composed of the Minuet and Fandango. The Menuet Allmandado is intermixed with steps from the Minuet and Allemanda. Other dances as well would merge and later these mixings would be called the Half and Half.

      The Modern term "Half and Half" as a dance was named in 1912 by Vernon Castle. It is basically a comedy routine whereas the dancers costume is half male and half female and the dancer would dance by/with themselves, miming the effect of dancing with oneself. The Cinquante-Cinquante which was invented by Mae Murray and Jack Mason in 1914 was a variation of the Castle's Half and Half.

      The costumes were originally half a tuxedo (and top hat) sewn together with the other half being a ball or cocktail gown or other costume

representing that dance style. The actual "Half and Half" dance was done to Waltz Music, but realistically could be any partner dance or Music.

     Burlesque and Vaudeville would soon start playing with this 'Half-and-Half idea with many female dancers designing whole skits/acts around this idea.

    ... Just a note:
     People have always been mixing dances together, whether because of lacking in knowledge, such as being weaker in one dance and stronger in another or would mix dances just because it was fun or just plain out of boredom. Plus there was a huge need for the Dance Masters of America to create new dances which helped fuel this form of mixing dances for something new. Music changes can make a dancer go from one dance to another as well within the same song. The castles probably noticed this and capitalized on it or just thought they were original, who knows?,  but the costume idea was certainly an original one as well as dancing with oneself.