I created a random isometry $T$ of $\mathbb{R}^3$ by generating a random orthogonal matrix $M$, uniformly distributed among all such, and a random displacement $v$, whose coordinates are drawn from a zero-mean normal distribution. For a point $p \in \mathbb{R}^3$, $T(p) = M . p + v$.

Let $T_1$ and $T_2$ be two such random isometries.
Now, starting with $p_0=(0,0,0)$, I formed the sequence of points
$p_1 = T_1(p_0)$,
$p_2 = T_2(p_1)$,
$p_3 = T_1(p_2)$,
$p_4 = T_2(p_3)$,
and so on, altering the two isometries one after the other, so
$$p_{2i} = T_2 \cdot T_1 \cdot T_2 \cdot T_1 \cdot T_2 \cdot \ldots \cdot T_1 ( p_0 ) \;.$$
Because the isometries are random, and include a random translation,
I expected the $p_i$ to wander off to infinity, and indeed this is common:

Above, the red segments connect $p_i$ to $p_{i+1}$.
Essentially these look like two spirals of points forming an unbounded `V`

.

However, (very) roughly 40% of the time, instead
spirograph-like bounded rings are formed
(or at least: apparently bounded, apparently rings):

So far inspecting the transformations numerically has not led me to
an understanding of this phenomenon.
If anyone sees intuitively why this process might lead to essentially
two qualitatively different shapes, I'd appreciate an explanation.
A specific question is:

Is there a positive probability that $|p_i|$ remains bounded as $i \to \infty$?

I thought the answer should be **No** but my simulations indicate **Yes**.

**Answered**. Both Benoît Kloeckner and Ofer Zeitouni answered the question, using rather different language but ultimately equivalently. One way to summarize the computational consequence their analyses is this: If $\det(M_1 \cdot M_2)=1$, the sequences is unbounded; if instead this determinant is $-1$, the sequence is bounded. As Benoît says, the analysis holds for any number of transformations ($k$-steps rather than two-steps), in which case the $\pm 1$ determinant of the product of the $k$ matrices signifies bounded/unbounded. Here is an example of a bounded 4-step sequence.

Now I can generate an infinite variety of these elegant figures! Thanks for everyone's help!