The twins corrected the mistakes on the test papers themselves.

我认为是做宾语，句子意思是这对双胞胎自己纠正了试卷中的错误．也就是by themselves．具体解释如下：
1）做宾语
a. 有些动词需有反身代词
absent, bathe, amuse, blame, dry, cut, enjoy, hurt, introduce, behave
We enjoyed ourselves very much last night. 我们昨晚玩得很开心。
Please help yourself to some fish. 请你随便吃点鱼。
b. 用于及物动词+宾语+介词
take pride in, be annoyed with, help oneself to sth.
I could not dress (myself) up at that time. 那个时候我不能打扮我自己。

注：有些动词后不跟反身代词， get up, sit-down, stand up, wake up等。
Please sit down. 请坐。

2） 作表语； 同位语
be oneself: I am not myself today. 我今天不舒服。
The thing itself is not important. 事情本身并不重要。

3） 在不强调的情况下，but, except, for 等介词后宾语用反身代词或人称代词宾格均可。如：
No one but myself (me) is hurt.