Can a confidence interval straddle the zero mark? From the section on Significance Testing in onlinestatbook:

Whenever an effect is significant, all values in the confidence interval will be on the same side of zero (either all positive or all negative).

Why? Can't a confidence interval be positioned both sides of the zero? Say, can't its range be [-12; +12]?
 A: 
Can't a confidence interval be positioned both sides of the zero? Say, can't its range be [-12; +12]?

Certainly it can. 
Did you miss the stated condition "Whenever an effect is significant"?
A: Often the quantity you are interested in has as its null value zero. So for example if you are estimating differences between means then the null value is zero - no difference. In such a case you would be interested in whether the confidence interval included zero. In some cases though the null value is not zero but perhaps unity. For instance if you are interested in the odds ratio then the null value is unity (since it is a ratio and when the quantities are the same their ratio is unity). In such cases you are interested in whether the interval includes unity and in fact the ratio cannot be negative.
A: The quote talks about testing whether a certain value is significantly different from zero at some level of significance.
If you had a confidence interval proposed by you, then the test would not reject the null hypothesis that "the tested value is zero". Because the interval [-12, 12] actually tells you that it might easily happen, that the true value is indeed zero.
A: Allow me to help. Basically, the confidence interval of a parameter are the endpoints of an interval or range that the parameter can reasonably achieve. So, if 95% of the time a parameter is greater than -12 and less than 12, then it could also be zero. And, if it can be zero then it can be worthless. For example, if $A X$ can have $A=0$ then its contribution is not significant to a regression. If we had 95% CI's for $A$ of 12 to 24, then it is not likely to be worthless as it is significantly not zero. Mind you, not significantly different from zero is not necessarily an insignificant contribution, and if we had more data, it might become significantly different from zero. It is a fine point, perhaps, that not significant does not mean insignificant, and just because we do not obtain significance as a result proves nothing special.
However, that does not mean that because we have a not significant result that it is also meaningless. It does have a meaningful context, and indeed one in which the confidence intervals contribute more than a probability alone would. 
Suppose we have 95% confidence intervals for $A$ as above that only extend from -1 to 1. Then they are 12 times lesser in range than the -12 to 12 CIs we had before, above. What that implies is that we would then know that a result of -5 or +5 would be significantly non-zero so that we now have a greater certainty of where the values of $A$ are not-located, and a narrower range of where our uncertain values of $A$ reside.
