The past can't be changed and neither should it. 

句子正确。这是并列句中，承前省略了重复的成分。

我猜测，网友见到的这类倒装句，多为不同的主语。主语相同的情况见的可能比较少，所以有疑问。本句 the past 和 it 指相同的主语。

= The past can't be changed and neither should it be changed.

我再举几个主语相同的句子：

Tom doesn't know the truth and neither does he want to know. 汤姆不知道真相，他也不想知道。

I don't like it, and neither do I care. 我不喜欢它，也不在意。

She hasn't seen me and neither has she heard from me. 她没有见过我，也没有收到过我的信。