The British Legion was founded in 1921 as a voice for the ex-service community as a bringing together of four organisations: the Comrades of the Great War, the National Association of Discharged Sailors and Soldiers and the National Federation of Discharged and Demobilized Sailors and Soldiers, and incorporated
the fundraising department of the Officers' Association.
The Officers' Association is still going today :
https://www.officersassociation.org.uk/our-history/
Regarding the badge , probably like other organisations of the time , when someone applied (and paid) for membership they would be given a badge.
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