According to Lee, Masuda and Park (page 3), the following result is "well-known in toric topology". I've found a proof, but I would like a published reference.
Let $X$ be a toric variety. Then the following are equivalent:
(1) $X$ is smooth and the poset of torus orbits (under closure) is the face lattice of the $n$-dimensional cube.
(2) There is a sequence of toric varieties and toric maps $X = X_n \to X_{n-1} \to X_{n-2} \to \cdots \to X_1 \to X_0 = \{ \text{point} \}$ such that each $X_i \to X_{i-1}$ is a $\mathbb{P}^1$ bundle. This is often called a "Bott tower".
The implication $(2) \implies (1)$ is clear. I can do the converse as well -- it's a clever bit of linear algebra -- but I'd rather just cite it. Does anyone know where this appears?
For the record, here is the combinatorially equivalent statement: Assume (1). Let $\Sigma$ be the corresponding fan, so $\Sigma$ is combinatorially the dual fan to a cube. In particular, it has $2n$ rays; call these rays $v_1^0$, $v_1^1$, $v_2^0$, $v_2^1$, ..., $v_n^0$, $v_n^1$ with $v_j^0$ and $v_j^1$ dual to antipodal faces. So the maximal cones of this fan are of the form $\text{Span}(v_1^{r_1}, v_2^{r_2}, ..., v_n^{r_n})$.
The condition that these cones form a fan says that $\det(v_1^{r_1}, v_2^{r_2}, \ldots, v_n^{r_n})$ has sign $(-1)^{\sum r_j}$, and the condition that the toric variety is smooth says that this determinant is $\pm 1$, so we can combine these as
(1') $\det(v_1^{r_1}, v_2^{r_2}, \ldots, v_n^{r_n}) = (-1)^{\sum r_j}$.
Meanwhile, (2) is equivalent to
(2') We can reorder the lower subscripts such that $\dim \text{Span}(v_1^0, v_1^1, \ldots, v_j^0, v_j^1) = j$, and $v_{j+1}^0+v_{j+1}^1 \in \text{Span}(v_1^0, v_1^1, \ldots, v_j^0, v_j^1)$.
The implication (1') to (2') is what I mean by "a clever bit of linear algebra"; I'll write it up if no one gives me a reference.