What switches can robocopy use to preserve file and directory times on the target?

Take a look at the options for the /COPY:[copyflags] and /DCOPY switches.

As per the ROBOCOPY /? usage info:

/COPY:copyflag[s] :: what to COPY for files (default is /COPY:DAT).
                      (copyflags : D=Data, A=Attributes, T=Timestamps).
                      (S=Security=NTFS ACLs, O=Owner info, U=aUditing info).

/DCOPY:T :: COPY Directory Timestamps.

For example:

ROBOCOPY c:\src d:\dest /MIR /COPY:DT /DCOPY:T

Will copy all files and folders and preserve the date and time stamps.

ROBOCOPY c:\src d:\dest /MIR /COPY:DAT /DCOPY:T

Will copy all files and folders and preserve the date & time stamps and file attributes.

There is also another (and I believe deprecated?) switch /TIMFIX which does much the same as /COPY:DT but it doesn't fix the time stamps on folders.

These were tested with ROBOCOPY 5.1.10.1027 on Windows 7 x64 Ultimate.

Be aware that the /MIR switch mirrors the directory that you are copying from; that is, /MIR will also delete files in the destination folder not found in the source folder. The /MIR switch is the equivalent of /E and the /PURGE switches used together (but with a minor exception).