I need to be able to do a periodic sync of content off my Windows 7 x64 laptop to my offsite CentOS 6 webserver.
Is there a tool/mode like rysnc for WinSCP or similar?
For a variety or reasons, Dropbox is not an option here.
Use Synchronize (Ctrl-S):

The prompts afterwards walk you through what you need to do.
Seems that WinSCP Isse 141 - Support for rsync will never come unfortunately.
I have a use case for downloading several Gigabytes from a Linux build machine to my Windows PC quite often. I managed to get this done with WinSCP and rsync using a Custom command in WinSCP.
Precondition: Windows For Linux (WSL) 2 on the Windows PC.
This command entered into WinSCP -> Preferences -> Commands -> Custom commands "Add..." -> "Add Custom command" will download all files (not recursively!) from the remote path to the current WinSCP local path on the Windows PC and show the progress in the console window that opens while the Custom command is executed by WinSCP:
bash.exe -c 'LOCAL_PATH=$(wslpath -a "!\"); rsync --progress "!U"@"!@":"!/"* ${LOCAL_PATH}; read -p "Press any key to continue..."'
Explanation
bash.exe is the WSL2 bash
wslpath is a tool in WSL2 that allows to convert a Windows path to the corresponding WSL2 based path.
!\ is WinSCP current local path. The path never ends with backslash.
!U is WinSCP current session username.
!@ is WinSCP current session hostname.
!/ is WinSCP current remote path. The path always ends with slash.
LOCAL_PATH=$(wslpath -a "!\") converts the WinSCP current local path to the corresponding WSL2 based path and stores it in a shell variable LOCAL_PATH.
rsync --progress "!U"@"!@":"!/"* ${LOCAL_PATH} is the WSL2 rsync command to download * files from username@hostname:remotepath/ to the WSL2 LOCAL_PATH with showing progress.
read -p "Press any key to continue..." will allow you to see the command line window and waits for a key press. Then the command line window will close.
There is cwRsync which I use for backing up data from Windows PCs to Linux servers.
Older versions are available without fee
As per @warren's answer, WinSCP can synchronize files.
If you want to do it periodically, you need to script it (WinSCP supports scripting) and schedule the WinSCP script using the Windows Scheduler.
The Sync-Option didn't fit my needs. cwRsync also didn't work for me, gave me an Error on Win7 x64.
I used cygwin's rsync (in a Cygwin-Shell) which worked fine. http://www.cygwin.com/