Lemma: Suppose $S=\{s_1, s_2, \dots, s_n\}\subset\mathbb Z$ where $2\nmid n$, then $$\sum_{1\leq i<j\leq n}|s_i-s_j|\equiv0\pmod2$$ Proof: WLOG suppose $s_1<s_2<\cdots<s_n$. $\sum_{1\leq i<j\leq n}(s_j-s_i)=\sum_{j=1}^n(j-1)s_j-\sum_{i=1}^n(n-i)s_i=\sum_{i=1}^n(2i-n-1)s_i\equiv\sum_{i=1}^n0s_i\equiv0\pmod2$

WLOG suppose the $2022$ marked points are $0, 1, \dots, 2021$. Let $A$ be the set of red points, $B$ be the set of blue points. By the lemma: The sum of the lengths of all the segments with any endpoints is $\sum_{0\leq i<j\leq2021}j-i=\sum_{i=1}^{2021}i+\sum_{1\leq i<j\leq2021}j-i\equiv\sum_{i=1}^{2021}i\equiv1\pmod2$. The sum of the lengths of all the segments with both endpoints red is $\sum_{i, j\in A,\ i<j}j-i\equiv0\pmod2$. The sum of the lengths of all the segments with both endpoints blue is $\sum_{i, j\in B,\ i<j}j-i\equiv0\pmod2$. $\Rightarrow$ the sum of the lengths of all the segments with a red left endpoint and a blue right endpoint $+$ the sum of the lengths of all the segments with a blue left endpoint and a red right endpoint $\equiv1-0-0\equiv1\pmod2$. $\therefore$ the sum of the lengths of all the segments with a red left endpoint and a blue right endpoint can't be equal to the sum of the lengths of all the segments with a blue left endpoint and a red right endpoint since both of them are integers.