  A matrix $A$ is totally positive (or non-negative) of order $k$, denoted
$TP_k$ (or $TN_k$), if all minors of size $\leq k$ are positive (or
non-negative). It is well-known that such matrices are characterized by the
variation diminishing property together with the sign non-reversal property. We
do away with the former, and show that $A$ is $TP_k$ if and only if every
submatrix formed from at most $k$ consecutive rows and columns has the sign
non-reversal property. In fact this can be strengthened to only consider test
vectors in $\mathbb{R}^k$ with alternating signs. We also show a similar
characterization for all $TN_k$ matrices - more strongly, both of these
characterizations use a single vector (with alternating signs) for each square
submatrix. These characterizations are novel, and similar in spirit to the
fundamental results characterizing $TP$ matrices by Gantmacher-Krein [Compos.
Math. 1937] and $P$-matrices by Gale-Nikaido [Math. Ann. 1965].
  As an application, we study the interval hull $\mathbb{I}(A,B)$ of two $m
\times n$ matrices $A=(a_{ij})$ and $B = (b_{ij})$. This is the collection of
$C \in \mathbb{R}^{m \times n}$ such that each $c_{ij}$ is between $a_{ij}$ and
$b_{ij}$. Using the sign non-reversal property, we identify a two-element
subset of $\mathbb{I}(A,B)$ that detects the $TP_k$ property for all of
$\mathbb{I}(A,B)$ for arbitrary $k \geq 1$. In particular, this provides a test
for total positivity (of any order), simultaneously for an entire class of
rectangular matrices. In parallel, we also provide a finite set to test the
total non-negativity (of any order) of an interval hull $\mathbb{I}(A,B)$.
