Do all modes make another mode when mirrored (i.e., intervals reversed)?

You're overthinking it. What composer would care what something like that looks like?

Concentrate on using your ears. The sound of the mode comes from the intended tonic by the composer. If I take C Ionian (Major) and emphasize the second degree (D) in a melody, then I'm playing Dorian. If I emphasize E, I'm playing Phygian and so on through the seven modes.

The harmony that the scale would be played over usually dictates the mode. So, mirroring the mode has no use. Basic theory and using ability to hear the sounds seems more logical than trying to come up with some other explanation for scale relationships.

A mnemnonic for which mode turns into which is numbering them 1-7 but starting from Aeolian rather than Ionian. Every mode reversed is the mode which brings the sum of the two up to 8. So Aeolian (1) is Mixolydian (7) reversed, Locrian (2) is Lydian (6), Ionian (3) is Phrygian (5), Dorian (4) is Dorian (4).

As this numbering makes A the first and G the last, I suspect this relationship may have been a factor in naming the tones.

Mirror Scale

This is called a mirror scale:

        ^{http://decipheringmusictheory.com/wp-content/uploads/2014/07/Mirrored-scales.png}

This form of mirroring doesn't have any underlying harmonic significance. For example, the ionian mode doesn't have any unique harmonic similarity or special relationship with its mirror, phrygian. The diatonic scale is built from a symmetric pattern of half/whole steps (which is easiest to see with the dorian mode, WHWWWHW). However, there is another type of mirroring that does have harmonic significance, and it's called "negative harmony."

Negative Harmony

Instead of making a single note the reflection point, we can use the range of notes extending from C to G as the reflection point.

B is one semitone below C. So to reflect the B, we look one semitone above G, which is A♭. A is three semitones below C. To reflect the A, we look three semitones above G, which is B♭. Using this method, we get these reflections, counting in semitones (S):

      B (1 S below C) → A♭ (1 S above G)
      A (3 S below C) → B♭ (3 S above G)
      G (5 S below C) → C (5 S above G)
      F (7 S below C) → D (7 S above G)
      E (8 S below C) → E♭ (8 S above G)
      D (10 S below C) → F (10 S above G)
      C (12 S below C) → G (12 S above G)

There is a special harmonic relationship between the original harmony and the mirrored harmony. Take the progression | A7 | D7 | G7 | CMaj |. Replacing the first three bars with their negative harmony counterparts, we get | E♭m6 | B♭m6 | Fm6 | CMaj |. In the original progression, the voice leading tended to resolve down. For example, a common voice leading line for the original progression is G → F♯ → F → E. Using the negative harmony, this line would now be C → D♭ → D → E. And instead of approaching CMaj by moving up fourths, we're approaching it by moving up fifths.

This type of negative harmony does have a lot of interesting and valuable meaning. The idea is attributed to Ernst Levy (see A Theory of Harmony). This example I've used is found in this interview with Jacob Collier:

I think this is just the result of the pattern that makes up all of our diatonic scales/modes. Since they all have the same set of intervals, just rearranged, patterns from one should appear elsewhere. In this case, I'd say you're somewhat arbitrarily rearranging intervals and finding that pattern in a different way than you usually would.

Since there are 7 notes in a diatonic scale and a defined set of intervals, the possibilities of what you can get when reversing the order are very limited. It might be easier to see this if you write out the intervals forward and backward over more than an octave.

W-W-H-W-W-W-H-W-W-H-W-W-W-H-W-W-H-W-W-W-H

H-W-W-W-H-W-W-H-W-W-W-H-W-W-H-W-W-W-H-W-W

That's the major scale intervals forward and backward. Below, I have placed some brackets to show the pattern.

(W-W-H-W-W-W-H-)(W-W-H-W-W-W-H-)(W-W-H-W-W-W-H)

H-W-W-W-H-(W-W-H-W-W-W-H-)(W-W-H-W-W-W-H-)W-W

Notice how the major scale is still in the reverse section, just displaced. This shows that when you reverse the order of the intervals, you will find the same scale in there somewhere. Since all of the modes are made up of these same intervals, it follows that no matter which mode you start with, you will end up finding another one when you reverse the order.

I'd also say that this shouldn't really point to some broader connection beyond it being a pattern. When I say that you're rearranging things somewhat arbitrarily, I intend to say that taking the intervals and going in reverse isn't really something that we associate with being a different key or tonality, eg, it's still the major scale when you play it going back down the octave.

Having said all of that, I'm hopeful that this isn't discouraging. Finding this pattern may not have been relevant in the sense that you were thinking but in finding it, you've come to better understand some aspects of theory, at the very least, making you look deeper into the modes. This curiosity is key in finding enjoyment in theory, which many are not able. I personally have had a whole lot of experiences like this, thinking I had stumbled onto something remarkable and realizing it's not as great as I thought, and not getting upset over that has allowed me to continue pursuing theory as a passion. Deep analysis of things that others gloss over as simple constructs, like Modes in the general sense, is what can bring you to the next place that may actually be a true revelation for you, if not everyone.

It is fairly easy to see that what you discovered always "works" for any mode.

If you put a mirror at one end of a keyboard, the reflection in the mirror "looks the same" in the sense that there are still alternating groups of 2 and 3 black keys in between the white keys. Therefore, if you count the whole-tone and half-tone intervals going down the scale from any white key, you get the same result as going up the scale from the mirror image of that key.

But of course the mirror image of a white note on the keyboard is a different note, except for D, so examples like "getting a locrian sound out of a lydian scale" don't seem to make sense.

But the Dorian mode is a special case, and that might have some musical implications for writing invertible counterpoint - there might be a musicological research project to be done starting from that observation....

A mode can be defined as a sequence of intervals. Reversing this sequence happens to result in another mode. To be specific: Ionian -> Phrygian; Dorian -> Dorian; Lydian -> Locrian; Mixolydian -> Aeolian. There's probably some mathematical reason to do with the relationship between modes (every mode is a rotation of the same sequence), and the fact that Dorian is symmetrical. But I'm not going to attempt to derive that.

So, can you do anything useful with this? I'm not so sure. To make it sound Locrian, you'll have to shift the root. In other words, F Lydian has the same notes as B Locrian, but you're not going to hear Locrian if the root stays around F. This is just standard modes stuff.

So, it's interesting, but I'm not sure it's musically useful. Please go and make something cool to prove me wrong. I promise I won't mind!