If I change the original to CMYK, I get the dull colors as in the picture in the OP, but if I then change back to RGB, the colors change but not back to the original?

Think of it this way:

When you change from RGB to CMYK, for each color, you are changing three numbers to four numbers. There is a plethora of ways to do that. For example, the product of 2*3*4 (24) can be expressed as 1*2*3*4, as 2*2*3*2, as 2*6*1*2, as 4*5*2*1.6666, or...

So you change a three component value (RGB) to a four component value (CMYK) according to any of a plethora of algorithms for doing that. Now you have a new CMYK value; there is no record of where that CMYK value came from. Then you change the four component value (CMYK) "back" to a three component value (RGB) according to any of a plethora of algorithms for doing that. But there is no "back" to it, because there is no single 1:1 correspondence between an RGB value and a CMYK value.

JET

Sinner-PWA wrote:

 

...

One point I will mention though.  If I change the original to CMYK, I get the dull colors as in the picture in the OP, but if I then change back to RGB, the colors change but not back to the original?

It is color matching and not color restoration which is impossible. When you convert colors from a wider color space like sRGB to a narrower color space like U.S. Web Coated (SWOP)v2, the color management tries to match the colors as much as possible but when the narrower color space doesn't contain the original colors they are clipped and this results in a color change. Then when the image is converted to a wider color space, if it encompasses all the colors of the image, you should see almost no color change except for some rounding of the values. Try changing the RGB color space in your Color Settings to AdobeRGB or ProPhoto and you should see no change in color appearance when converting from CMYK to RGB. The default sRGB color space has relatively narrower gamut and some parts of the blues and greens of the CMYK color space stick out of the sRGB color space because they are not fully encompassed from the sRGB color space.

You can have a very good idea about the different color spaces and compare them in a 3D plot if you check this site: http://www.iccview.de/content/view/3/7/lang,en/  you may need to install the 3D viewer plug in for your browser in order to view the 3D comparisons. And don't forget to drag you mouse to see the color spaces in the 3D plot from all sides. Holding Shift will rotate the view faster. Examining color spaces in 3D tells the whole story while 2D charts don't tell much.

For example here's a screen shot from an angle that reveals the parts of the CMYK space that stick out of the sRGB color space. When the image has colors in that parts for example pure cyan, the color appearance will shift when converted to sRGB. but will remain the same if you convet to AdobeRGB or ProPhoto.