The reality (as in physical reality, the phenomenon) is that a pixel's "value" is determined both by what is happening along the X dimension and the Y dimension (in k-space).
If you want to reconstruct an image you have to do it from **two spatial sinusoidal waves.
This is represented in the $f[m,n] \cdot e^{-j 2 \pi (u m + v n)}$ part of the DFT. This is the product that we sum along the $u$ and $v$ directions.
Notice here, that to obtain one $u,v$ value you need to evaluate sinusoids in both the $u$ and $v$ directions. And vice-versa of course, meaning that the grey level value of one pixel is decomposed in the coefficients for both sinusoids along the $u$ and $v$ directions. If you only run one of them, you get only HALF the story.
If you reconstruct an image from rows, you synthesize grey level variation from just one direction. You know how a pixel's value varies with respect to its left and right neighbours but not its top and bottom neighbours.
Here is a mental experiment: Take an image and run DFTs along the rows (that is, the horizontal direction, as per the recipe that motivated this question). Now take the original image and add 42 to the pixels of the rows of the upper half (this looks like a step in the vertical direction). What is the effect of this? You are only introducing a DC to the ROW DFTS, other than this, the rest of the spectrum is exactly the same.
You can choose to get even more adventurous in that vertical direction and modulate the pixels by sinusoids. They will go completely amiss, why?
Because that modulation, along the vertical direction only introduces some "disturbance" to the DC component of the horizontal direction. It is impossible to pick up anything else unless you "check" for it, by evaluating the DFT along the vertical dimension too.
And you can see this happening in $F[u,v] = \sum_m \sum_n x[m,n] e^{-j 2 \pi (u*m + v*n)}$ because the sums are nested as well as when you apply the DFT twice because you first apply it on the rows (now you know how a pixel varies with respect to its left-right neighbours) and then you apply it along the columns of the ROWS DFT (now you know how a pixel varies with respect to its top and bottom neighbours).
Hope this (and to an extent this) helps.