How can you do two-way Dropbox <--> local folder syncing on iOS?

Unfortunately, I'm almost certain there's no equivalent Dropbox solution for this currently on iOS.

While I cannot give you a technical reason why two-way syncing on iOS is not possible, I can say that at a high level, the iOS platform is more guarded in terms of allowing developers to access certain capabilities of the device.

As you stated in your question, Android devices are capable of two-way synchronization through apps like Dropsync; this is because of the relatively more open-ended nature of the Android platform for both developers and users. To read more on why this is the case, read the 'Android Offers an Open Platform' section of this article.

To confirm, I contacted two customer representatives of two data-sync companies: Loom and SugarSync. I asked both reps the following question:

Is possible for two-way synchronization of files between an iOS device and a computer? For example, if I sync pix from my iPhone to my iMac, and then edit/delete the pix on my computer, will the changes be reflected automatically on my iPhone?

Both times, the answer was:

No, this is not possible due to the closed nature of the iOS platform.

If this capability is very important for your work, I would suggest migrating back to an Android device.

If you want/need to use your iPhone for this, and if you need a platform-independent solution (i.e., you don't want to use iCloud), I'm afraid tedium is in your future. I would suggest that you keep a list of the photos as you edit/delete them on you computer, so that you know which photos have been updated and need to be manually re-download from Dropbox (or manually deleted) on your iPhone when you use it next.

If you're willing to jailbreak, you can use iFile to two-way sync a folder in /var/mobile with a folder in Dropbox. It can't do background two-way sync, but you can do it manually.

I know Documents by Readle supports two-way sync with Dropbox and other cloud services on iOS. I use it all the time sync books from my computer and iOS devices.