Members of the Allgemine SS (General SS), primarily guarded concentration camps.
Incorrect. "Allgemeine SS" is an imprecise term used to mean the general/main/primary SS organization, itself subdivided into its 8+ departments and 150+ sub-departments and/or the volunteer paramilitary groups in the Reich. After the war SS veterans used it as a means of saying "the other guys ran the camps, not me."
From August 1940 to March 1942, concentration camps were "Department Six" (Amt VI) of the SS-FHA (main operations office) - the same office that managed the Waffen-SS.
From April 1941 on Concentration camp guards were part of the Waffen-SS, wearing Waffen-SS uniforms and carrying Waffen-SS paybooks.
In March 1942 concentration camps administration was reorganized under Department Group D of the SS Economic & Administrative Office (SS-WVHA).
They were recruited essentialy for this purpose.
The SSTK was founded to guard the KZ (concentration camp) installations.
In December 1939 Himmler transferred 6,000 "Allgemeine-SS" reservists (meaning the guys in the regionally located, volunteer SS battalions all over Germany) into the Totenkopfwachsturmbanne (concentration camp guard units)
In the summer of 1944 Wehrmacht personnel (many of them Luftwaffe) were transferred into the Totenkopfwachsturmbanne for concentration camp and slave labor guard duty. Many of these troops wore the "double swastika" collar tab,
Waffen-SS troops of all divisions were rotated in on occasions for duty at these centers.
Basically correct, although the transferrs were done on an individual as opposed to a unit level. And you wouldn't necessaryily be transferred to "concentration camp duty" - you'd be assigned to SS-WVHA, Amtg D, stationed at Dachau.
Almost all SS departments were involved in the concentration camps. It was the nature of the system.