This is a Ceilidh style dance, of "traditional" origin. It is known under a number of names with no real "original" description available.

A description of this dance was published end of 1939 in the magazine "English Dance and Song" as "The Square Eight" with a subtitle "as danced in Cumberland, near the Scottish Border". You may want to read the scans at <<http://www.contrafusion.co.uk/Dances/EFDS3911-CumberlandSquareEight.html|contrafusion>> for more information.

John Keppie wrote it up in 1992 as "Cumberland Square" in the book "Burns Night in the Annapolis Valley" and states "This is a North Country dance arranged for Scottish Country dancing. Square set of 4 couples with sets lined up across and up-and-down the room so that the sets can intermingle during bars 1-16".  Danced 2 x 64.

More recently in 2017 a dance description entitled "The Cumberland Square Eight" was published in the "Collins Little Book: Scottish Dance".

The dance has also been described as "The Cumberland Square Eight" in the <<http://www.scottishdance.net/ceilidh/dances.html#CumberlandSquareEight|Ceilidh Dance instructions>> on the Grand Chain server.

Furthermore, "The Cumberland Square Eight" is mentioned in a number of books and novels such as "Disaster with the Fiend" by Sheila Lavelle (1988, p.76-83), "Popular Leisure in the Lake Counties" by Lyn Murfin (1990, p.194) or "Annie's Girl" by Audrey Howard (2002).

From the published names one tends to speculate that while "The Square Eight" might have been an original name, the term "Cumberland" might have been added to distinguish this version danced in Cumberland (near the Scottish border) from other dances like the "Goathland Square Eight" or the "Yorkshire Square Eight". The abbreviation to "Cumberland Square" may be misleading as judging from the above sources it seems unlikely that this dance refers to a location called Cumberland Square.

In <<https://my.strathspey.org/dd/issue/2633/|issue 2633>> Meryl Thomson provides information about the source of the dance:

The dance "Cumberland Quadrille" appears in the handwritten notes of Thomas Murphy c.1830s, now held in the Mitchell LIbrary, Glasgow. The instructions appear to be a precursor of the current dance - Top and Bottom Highland Scottische poussette to opposite places. Sides the same. Top and Bottom repeat Back, Sides repeat back. Top and Bottom rights and lefts turn partners sides Ladies Chain all set to corners, Balance & turn partners, grand Chain half round, promenade to places all waltz.

This was a single quadrille - quadrille being a historic term for a dance for 4 couples in a square set. 

As for the "Basket swing" formation danced by 4 people in the center of the set - there are two versions mentioned as alternatives in e-crib. Both can be seen in videos of this dance.