We introduce a class of metrics on
R
n
\mathbb {R}^n
generalizing the classical Grushin plane. These are length metrics defined by the line element
d
s
=
d
E
(
⋅
,
Y
)
−
β
d
s
E
ds = d_E(\cdot ,Y)^{-\beta }ds_E
for a closed nonempty subset
Y
⊂
R
n
Y \subset \mathbb {R}^n
and
β
∈
[
0
,
1
)
\beta \in [0,1)
. We prove, assuming a Hölder condition on the metric, that these spaces are quasisymmetrically equivalent to
R
n
\mathbb {R}^n
and can be embedded in some larger Euclidean space under a bi-Lipschitz map. Our main tool is an embedding characterization due to Seo, which we strengthen by removing the hypothesis of uniform perfectness. In the two-dimensional case, we give another proof of bi-Lipschitz embeddability based on growth bounds on sectional curvature.