Many things are possible. Installing an SSH Server app (e.g. SSHelper or DigiSSHD / SSH Server) would give you the possibility to login via ssh, or remote-copy via scp.
Terminal IDE (available only for Android 4.4 and earlier, according to the app description) gives you a quite complete package, even including the famous Midnight Commander, and a lot more ...
You would use adb push --sync, for example:
adb push --sync /home/abc/ /sdcard/abc
from adb --help:
push [--sync] LOCAL... REMOTE
copy local files/directories to device
--sync: only push files that are newer on the host than the device
but it seems that this function is available for relative newer ADB version such as version 1....
As we figured out the solution you now use in chat, this answer should comply with "being tested", "not involving a PC or file manager" and "relying on verified sources". Though in the meantime there was another rsync answer posted, you've asked me to present our solution as well – so here we go:
The rsync command can be used to accomplish that. Taken all ...
The culprit is android's crippled ls command. Zsh's completion run ls -d1FL which /system/bin/ls doesn't understand.
A way to fix this is to edit /data/data/com.arachnoid.sshelper/home/.ssh/environment such that /data/user/0/com.arachnoid.sshelper/bin is the first entry in PATH. (SSHelper ships with a more complete ls)
Note that SSHelper might rewrite ...
First on Android each app has an own user-id/account, hence u0_a312is the user of the app that runs the ssh server. Regular Linux user accounts like myuser don't exist on Android.
Based on the path /storage/6632-6330/ I would assume that you are trying to access the external sd-card.
On a non-rooted device write access to ext-sd is not possible on Android ...
While this answer is not exactly an explanation on the usage of adb sync, it may help when your primary goal is to sync files using adb.
You may be interested in adb-sync which is a Python script that uses adb to synchronize files from your computer to your Android device or vice versa.
Example (computer -> device):
adb-sync /home/abc/ /sdcard/abc
If you paid for the Pro version of Titanium Backup, you should have the option to store the backups to several online storage locations. Google Drive, Dropbox, and Box are built-in to Titanium Backup Pro. All you need to do is add your account info, then tell your schedule to back up to those locations.
This is how my phone is set up. If I lost my phone, I ...
Your SD card contains pretty much all the data you need.
In addition to app backup, you can also use SMS Backup+ to backup your texts to the SD card (and then to the home server).
Titanium Backup Pro version has an option to backup files to your Dropbox folder, so you can use that instead of rsync if you like.
Also, try to move as much data to the cloud ...
"adb sync" is not relevant in your case. It is used to sync system partition to your connected adb device. It will only work if your phone is rooted and you first need to put system partition into writeable mode. By default it is in read-only mode. Please follow this article for more details: