When are two neighbouring fractions in Farey sequence are similarly ordered

$a_{i + 1}b_i - a_ib_{i + 1} = 1$ is indeed the right identity to look at.

Note that, since $\frac{a_i}{b_i} < \frac{a_{i + 1}}{b_{i + 1}}$, the only way $(a_{i + 1} - a_i)(b_{i + 1} - b_i) \ge 0$ could possibly fail is if $a_{i + 1} \ge a_i + 1$ and $b_{i + 1} \le b_i - 1$. But then we'd have $$a_{i + 1}b_i - a_ib_{i + 1} \ge (a_i + 1)b_i - a_i(b_i - 1) \ge a_i + b_i > 1$$ which is a contradiction.

The non-negative Farey fractions can be obtained by starting with $\frac01,\frac11,\frac21,\ldots$ (where the claim clearly holds) and then repeatedly inserting $\frac{a+c}{b+d}$ between adjacent fractions $\frac ab$ and $\frac cd$. Near an inserted fraction, we have $$ ((a+c)-a)\cdot ((b+d)-b)=cd\ge 0$$ and $$ (c-(a+c))\cdot (d-(b+d))=ab\ge0.$$