For a two-sample t-test (paired or unpaired), what you are looking at is the difference between the means of the two samples. The 95% confidence interval is providing a range that you are 95% confident the true difference in means falls in. Thus, the CI can include negative numbers, because the difference in means may be negative.
For a very basic example, let's say that your control group has a mean of $1$ and your treatment group has a mean of $2$. The difference between these will be $-1$. When you calculate the confidence interval for the true difference in means (not just the sample difference), it will be centered on $-1$. The confidence interval (whatever it is) will by definition fall more in the negative side than the positive side. However, if you reversed the calculation and did treatment-control instead, you would get a range falling more in the positive side. It would not affect your final conclusion.
[EDIT] The numbers in the question got updated, but I'll leave this comment here for future reference: In the numbers you give (as whuber points out in a comment), your confidence interval should center on $-0.033$ because that's the difference in your sample means. Because it doesn't, there's likely some error in your calculations somewhere.