Actually, it is not meant as an ironic counterpart, but as a worthy folk festival in honour of carpenters in saint San Giuseppe. Carpenters, because the city has a proud tradition of woodworking. It has organised the festival since 1966 and has built it up around many of the traditions of horse race in Siena, hence the city's eight districts that compete among themselves for the banner "il Palio".
The event runs from Sunday to Sunday sometime in mid-March and the end takes place the last weekend with several festivities. On Sundays it all ends with a suitable parade that goes through the city to the racecourse, while the voltage rises towards the great decisive race that takes place on the donkeys. It is recommended to attend the weekend through, which generally should be seen as an opportunity to taste traditional dishes, attend street entertainment and medieval performances.
The victory is celebrated in the winning neighbourhood with a banquet and festivities to late the evening. And everybody is invited.
The truth about donkey races is that they are very popular throughout Italy and therefore, it is not intended as an ironic counterpart to the horse races, but more as a cultural and historical tradition, as also seen in cities like: Alba in Piedmont (in connection with the autumn truffle festival) and Asciano in Tuscany, just to name a few.
For additional information about the donkey race in Torrita di Siena: http://www.paliodeisomari.it/