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The spherical Vostok capsule separated from the rest of the spacecraft when beginning descent. The center of gravity was "below" the cosmonaut's couch, so the capsule didn't tumble. But what prevented it from spinning? There was no parachute to do so; the cosmonaut always ejected and used his/her own parachute to land.

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    $\begingroup$ Remark: "It was possible to land, and without leaving the cab, in the descent vehicle, which descended at a speed of about 10 m / s.", "Можно было осуществлять посадку, и не покидая кабины, — в спускаемом аппарате, который опускался со скоростью около 10 м/с. " astronaut.ru/bookcase/books/popov02/text/05.htm $\endgroup$ – A. Rumlin Nov 29 '19 at 10:52
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The Vostok capsule wasn't stabilized during reentry. Friction from reentry was probably the major factor to not let it spin wildly.

I couldn't find information about other flights, but the first one (Vostok-1) is described with following phrases from Gagarin's debriefing:

Как только выключилась ТДУ, произошел резкий толчок, и корабль начал вращаться вокруг своих осей с очень большой скоростью. Земля у меня проходила во «взоре» сверху справа вниз и влево. Скорость вращения была градусов около 30 в секунду, не меньше... ...Разделение резко почувствовал. Произошел хлопок, затем толчок... Вращение шара продолжалось по всем осям с прежней скоростью (30° в секунду). Затем начал чувствовать торможение... ...Начинается замедление вращения корабля; причем по всем трем осям. Корабль начал колебаться примерно на 90° вправо и влево. Полного оборота не совершалось. По другой оси такие же колебательные движения с замедлением... ...Затем начался плавный рост перегрузок. Колебания шара все время продолжались по всем осям. К моменту достижения максимальных перегрузок я наблюдал все время Солнце. Оно попадало в кабину в отверстие иллюминатора люка № 1 или в правый иллюминатор. По «зайчикам» я мог определить примерно, как вращается корабль. К моменту максимальных перегрузок колебание корабля уменьшилось до ±15 градусов... ...Затем начался спад перегрузок... ...Когда я начал готовиться к катапультированию, корабль развернуло к Солнцу примерно на 90°.

Translation:

As soon as TDU (braking engine) shut down, there was a sharp jolt. The spacecraft started spinning about its axes with very high speed. The Earth was passing in the Vzor (window) from top-right to bottom-left. The speed of rotation was around 30 degrees per second, no less. ... [I] sharply felt separation [of the service module]. There was a clap (or bang, meaning sharp sound), then a jolt... The rotation of the ball [the reentry capsule] continued along all axes at the same rate (30° per second). Then [I] began to feel braking [meaning aerobraking, decceleration due to friction] ... ... The rate of the spacecraft rotation began to fall; and on all three axes. The capsule began to oscillate about 90° to the right and left. Rotation stopped going for the full circle [now] [meaning continuous rotation has stopped]. [I was observing] the slowing down of oscillatory movements along the other axis in the same manner... ... Then deceleration (g-force) started to grow slowly. Oscillations of the ball [the capsule] continued all this time along all axes. By the time I reached maximum g-force, I was observing the sun all the time. It fell into the cabin through the window of hatch No. 1 (into the window on [my] right). [Looking at] the sun beams I could determine approximately how the spacecraft rotates. By the time of maximum g-force, the spacecraft's oscillations decreased to ± 15 degrees ... ... Then g-force started to decrease ... ... When I began to prepare for ejection, the spacecraft rotated about 90° towards the sun.

So, the capsule was rotating as it pleased, basically. I guess slow rate rotation shouldn't have been a problem. Judging by the sketch below, uncontrolled (slow) rotation would only mean the cosmonaut was to land unpredictibly anywhere withing the landind radius (~2km) from the capsule. (Note the capsule had it's own drogue and main chutes, which deployed after cosmonaut's ejection)

enter image description here

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    $\begingroup$ Thank you for a very thorough answer. So, a rotation slow enough to be tolerable, at least by an experienced test pilot. No dangerous spin, and it didn't interfere with the ejection. $\endgroup$ – SpaceInMyHead Nov 30 '19 at 22:06

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