The Half and Half is a Ragtime dance done in 5/4 time.

According to the introduction of Modern Dancing :

The Half and Half is an original drawing-room dance invented by Mr. Castle. It combines the best steps of the Hesitation and the Maxixe, but the tempo is entirely new.
In the mid 19th century, the same dance (more or less) was called the 5/4 waltz or the waltz a cinque temps.

Mr. Castle calls it the half and half, because he liked to describe it as half waltz (three beats) and half One Step (two beats).

Personally, I'd just call it Fox Trot as Mr. Castle describes normal Fox Trot moves in 5/4 time with various combinations of quicks and slows. Actually, I've found it simpler to use the terms Quick (one beat) Slow (two beats) and Very Slow (three beats). This has an added benefit in that the phrase "Very Slow" has three syllables for three beats : handy when calling out the step.

The most common rhythm in Modern Dancing  is (to start on count one) Very Slow, Quick, Quick. He makes the dance more interesting by letting variations cross the measure, and so the same rhythm would be more often described as Quick, Quick, Very Slow, but starting on count four.

Generally, when taking a Slow of Very Slow step, the free leg is straight, and either striking a pose (arabesque) or making an arc (rond de jambe).

As with the Ragtime One Step, any combination of Quicks, Slows and Very Slows, mixed with any Step, Style and Hold is fair game.