How to use SIFT with a color inverted image

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Do you mean by "gradient direction" keypoint orientation on which descriptor orientation depends? What do you mean by "swap the contents"? We have 8 bins histogram for each 16 regions (8x4x4=128), you mean to switch values of the opposite bins? for example 1st and 5th, 2nd and 6th?

The gradient direction is the direction of the gradients as gradients are defined in their mathematical sense. This then applies to both the orientation assigned to the interest point as well as to the gradients that are interpolated and binned into the SIFT descriptor bins. By "swap the contents" I meant that you effectively have to swap bins. There is no need to recompute the SIFT descriptor a simple permutation of the already compute bins will do.

thanks for the explanation, but as you can see here i.stack.imgur.com/XRfdh.png gradients are not absolutely opposite

That looks odd indeed. I would first test this on a simple black and white image that only contains simple shapes as a circle, triangles and rectangles.

Looks the same.. http://i.stack.imgur.com/fY2PR.png But when I subtract 180 or add 180 (within [0, 360]) looks like http://i.stack.imgur.com/Oniad.png , interesting