I set up a temporary antenna yesterday and connected it to my rig,
including an MFJ Versa Tuner II (Model MFJ-949D), purchased about 15
years ago.

Setting the transmitter (Icom-725) to low output power (about 10W) I
closed the CW key and the tuner indicated about 15W of forward power,
and no reflected power. That is, it "indicated" a SWR of 1:1.
Unlikely, to say the least.

The reflected power meter just doesn't move at all. Regardless of band.
Thus, I am convinced that something is wrong with it (it has been
moved twice since purchase). I did open the case and nothing looks out
of order. Touching the reflected meter terminals with an ohm meter
causes the reflected meter needle to jump, so it appears that the meter
itself is working.

The antenna seems to work very well for receiving. And I am able to
"get out", as it were. I can hear myself on a little SW radio I have at
14.2 MHz in AM.

Questions:

Any thoughts as to where to look for the "problem" with the tuner?

Is there another way to determined SWR with just the meters on the
transmitter (RF power out) and the forward power meter on the tuner?

You can look at the power and then interchange the input and output of the
meter to see what the swr actually is when you are not using the switches in
the tuner position but in the bypass position.
Thank I would look at the diode first to see if it good. Being a MFJ you
may need to see if any of the connections are loose. I have one similar to
that tuner and the first thing I did was open it up and tighten up all the
loose screws. Bought It new at a hamfest from a dealer and when I got it
home I noticed it making a noise when moving it around. Found a loose nut
and washer inside it.