Photographing through a difraction grating resuts in a color blurring on the diffracted image which seems to destroy the image. It is my discovery (see references at footnote) that for each wavelength we have a different, rotated view of the object.
Before showing that with more complex figures, I will do it by means of a few light rays and the principle of inverse path of the light. If we can generate a luminous reconstruction performing the inverse path of the rays of an object  we get an image exactly as the original, that means also identical to a holographic image, but inverted in relief (named pseudoscopic, looking as the moldure of the object).

Let us see in figure a) the scheme of photographing an object through a diffraction grating.  We employ the law of diffraction through a grating in such a way to keep the properties of the result, even if the figure does not have the proportions of a real experience.

Of the white light traversing the grating, only two rays are represented, each with a different wavelength. Given the conditions of our experience, they are the only ones exiting from the specific object point chosen to reach the film at that position.

We assumed that the aperture of the photographic objective is very small, as if we were dealing with a pinhole camera.
In figure b) we have the film already developed and assume that he has the perfect condition of registerig and reproducing the wavelength of each ray. Now, he is transilluminated from behind with white light in all directions and we can see by analysis  that only two rays which are reciprocal to the two registered rays are the only ones to make the indicated path, inverse to that of registering.

We can obtain in that way the image of an object exactly at its original position, but can only be see for anobserver located at a position where he can be reached for that rays. That is, from the opposite side to the viewpoint of the camera (upper left side, in this case).
The observer sees the image inverted in depth (pseudoscopic). Fortunately, there are two simple ways to make it to be seen in natural relief: by inverting the photographic film or changing the viewing to the symmetric diffracted observed order.

In figure c) we inverted the film.  Observe carefully and you can verify that the ray that was below in  figure b) is above now and the their virtual prolongation makes it to reach the virtual prolongation of the other ray. We obtained a virtual image in normal relief, what can be cheked by analysing the symmetry of the situation for  any other object points.

This analysis is correct and we made experiences with ordinary color film obtaining results which are  visually approximated of our desired solution. We know that the result coul be perfect if the film could reproduce perfectly wavelengths.  Color film reproduces reasonable well the sensation of color, but not the wavelength. That is, the visual sensation, not the distribution of wavelengths. Fortunately in this case too, there is a tecnic which reproduce wavelengths, Lippman photography,  by G. Lippman, Nobel prize of Physics in 1908, already referred at the begining of this site on holography.
See a photograph of a Lippman photograph
Applying Lippman photography to our procedure we will have holograms registered under white light.  We are trying to reproduce those old techniques, not in the same way, but in such a way that could be easier to work and employing holographic film instead of trying to reproduce Lippman plates.

For a more complete description, I include below other schemes for the same technique but employing for projection a special diffracting screen I invented.

We see in 1) the position of the images of a cubic object at the registering stage.  Its color is directly related to its wavelength.

Due to the large focal depth we obtain the result as a compression wich puts the volume images on the plane of the film.

Some rays appear in figure 2) also with inverse direction, just to anticipate figure 3) where every ray have makes the path which is inverse to that of the corresponding rays of the registering. Those reciprocal rays are the only ones that can reach the center of the lens.

In figure 3) we have the situation of inverse path that, being exactly drawn, would show every ray going to the same point, the center of the photographic objective.

Finally, in the figure below, I put a more complete scheme, with some inclination of the film for appropiated focusing at the screen and where we can perceive that color reproduction can not be achieved, because the wavelength of each observed ray changes according to the observer's position.

References:
- "The encoding of depth by spectral diffraction", artigos básicos

- "New possibilities in the utilisation of holographic screens", Proc. of the SPIE meeting "Electronic Imaging", conference "Practical Holography VI", San Jose-CA-USA, 1992.02.9-14, p.289-293.