We prove that non-negative solutions to the fully anisotropic equation
∂
t
u
=
∑
i
=
1
N
∂
i
(
|
∂
i
u
|
p
i
−
2
∂
i
u
)
,
in
R
N
×
(
−
∞
,
T
)
,
\begin{equation*} \partial _t u= \sum _{i=1}^N \partial _i (|\partial _i u|^{p_i-2} \partial _i u), \quad \qquad \text {in } \mathbb {R}^N\times (-\infty , T), \end{equation*}
are constant if they satisfy a condition of finite speed of propagation and if they are both one-sided bounded, and bounded in
R
N
\mathbb {R}^N
at a single time level. A similar statement is valid when the bound is given at a single space point. As a general paradigm, local Hölder estimates provide the basics for rigidity. Finally, we show that recent intrinsic Harnack estimates can be improved to a Harnack inequality valid for non-intrinsic times. Locally, they are equivalent.