This page should really be titled "DDR marked headwear and its possible use by the Stasi."

According to the very knowledgeable people in the DDR sub-forum of Wehrmacht-Awards.com headwear marked "DDR" was used by the Stasi, but not exclusively by them. So the hats and caps on this page may be Stasi or could just be NVA (excluding the 1960 dated visor that is).To read the forum thread the link is below.

Wehrmacht-Awards.com

The 1960 dated visor cap shown on the left originally had the above embroidered bull's-eye which was cut off and replaced with the State emblem.

The original maroon piping was cut off and white piping sown on in its place, the thread is visible on the inner lining of the hat.

In 1961 the army as well as Stasi switched from unit specific piping on their headwear to white piping. The old Black-red-gold Bull's-eye cockade was used up until 1961 on officers headwear and as late as 1963 for enlisted men and was replaced by the circular compass and hammer cockade. The hand embroidered officer wreath was replaced by a two piece metal wreath cockade in 1965.

Ministerium fur Staatssicherheit-Stasi-MfS-Ministry for State Security.
Officer, silver hand embroidered wreath and metal cockade, silver cap cord. Originally this hat had an emdroidered bull's-eye cockade the remains are shown in image #1. The orignal dark red piping was mostly removed or trimmed down #2, and the white piping sown on and the white thread used can be seen inside the hat #3. This occurred around 1964/65 in accordance with new uniform regulations. Thanks to Tony for informing me of this. Size 58 and dated 1960. Brown band. (Visor type "Pre-65). VEB 1856. Marked DDR."
Officer, gabardine fabric, two piece cockade and silver cap cord , white piping. Size 54. Marked DDR, dated 1969. Brown band. (Visor type "4-U"). Veb 1856.
Officer, gabardine fabric, two piece cockade and silver cap cord, white piping. Size 54. Marked DDR, dated "X" (1971). Gray band. (Visor type S-2882 2 far 1, i-l). Veb 1856. 
Officer, gabardine fabric, two piece cockade and silver cap cord, green piping. Size 59. Marked DDR, dated "M" (1970). Brown band. (Visor type S-2882 2 far 1, i-l). Veb 1856.
DDR marked Officer Winter hat. Imitation fur, embroidered wreath metal cockade. Size 57, dated 1965
Early Issue Career soldier side cap. Gabardine fabric, textile circular cockade, no piping. Size  53 , no date. On the inside of the cap is stamped "DDR 53 1856 23171".