We study topological groups
G
G
for which either the universal minimal
G
G
-system
M
(
G
)
M(G)
or the universal irreducible affine
G
G
-system
I
A
(
G
)
I\!A(G)
is tame. We call such groups “intrinsically tame” and “convexly intrinsically tame”, respectively. These notions, which were introduced in [Ergodic theory and dynamical systems in their interactions with arithmetics and combinatorics, Springer, Cham, 2018, pp. 351–392], are generalized versions of extreme amenability and amenability, respectively. When
M
(
G
)
M(G)
, as a
G
G
-system, admits a circular order we say that
G
G
is intrinsically circularly ordered. This implies that
G
G
is intrinsically tame.
We show that given a circularly ordered set
X
∘
X_\circ
, any subgroup
G
≤
A
u
t
(
X
∘
)
G \leq {\mathrm {A}ut}\,(X_\circ )
whose action on
X
∘
X_\circ
is ultrahomogeneous, when equipped with the topology
τ
p
\tau _p
of pointwise convergence, is intrinsically circularly ordered. This result is a “circular” analog of Pestov’s result about the extreme amenability of ultrahomogeneous actions on linearly ordered sets by linear order preserving transformations. We also describe, for such groups
G
G
, the dynamics of the system
M
(
G
)
M(G)
, show that it is extremely proximal (whence
M
(
G
)
M(G)
coincides with the universal strongly proximal
G
G
-system), and deduce that the group
G
G
must contain a non-abelian free group.
In the case where
X
X
is countable, the corresponding Polish group of circular automorphisms
G
=
A
u
t
(
X
o
)
G={\mathrm {A}ut}\,(X_o)
admits a concrete description. Using the Kechris–Pestov–Todorcevic construction we show that
M
(
G
)
=
S
p
l
i
t
(
T
;
Q
∘
)
M(G)={\mathrm {Split}}(\mathbb {T};\mathbb {Q}_{\circ })
, a circularly ordered compact metric space (in fact, a Cantor set) obtained by splitting the rational points on the circle
T
\mathbb {T}
. We show also that
G
=
A
u
t
(
Q
∘
)
G={\mathrm {A}ut}\,(\mathbb {Q}_{\circ })
is Roelcke precompact, satisfies Kazhdan’s property
T
T
(using results of Evans–Tsankov), and has the automatic continuity property (using results of Rosendal–Solecki).