  For an $n$-bit positive integer $a$ written in binary as $$ a =
\sum_{j=0}^{n-1} \varepsilon_{j}(a) \,2^j $$ where, $\varepsilon_j(a) \in
\{0,1\}$, $j\in\{0, \ldots, n-1\}$, $\varepsilon_{n-1}(a)=1$, let us define $$
\overleftarrow{a} = \sum_{j=0}^{n-1} \varepsilon_j(a)\,2^{n-1-j}, $$ the
digital reversal of $a$. Also let $\mathcal{B}_n = \{2^{n-1}\leq a<2^n:~a
\text{ odd}\}.$ With a sieve argument, we obtain an upper bound of the expected
order of magnitude for the number of $p \in \mathcal{B}_n$ such that $p$ and
$\overleftarrow{p}$ are prime. We also prove that for sufficiently large $n$,
$$ \left|\{a \in \mathcal{B}_n:~ \max \{\Omega (a), \Omega
(\overleftarrow{a})\}\le 8 \}\right| \ge c\, \frac{2^n}{n^2}, $$ where
$\Omega(n)$ denotes the number of prime factors counted with multiplicity of
$n$ and $c > 0$ is an absolute constant. Finally, we provide an asymptotic
formula for the number of $n$-bit integers $a$ such that $a$ and
$\overleftarrow{a}$ are both squarefree. Our method leads us to provide various
estimates for the exponential sum $$ \sum_{a \in \mathcal{B}_n} \exp\left(2\pi
i (\alpha a + \vartheta \overleftarrow{a})\right) \quad(\alpha,\vartheta
\in\mathbb{R}). $$
