Haier are known to have had some quality problems, but it appears they may be solving (some of) them. Even so, it's a gamble.
Danby has three good ones at reasonable prices. One is a front-loader you can get at Home Depot for about $500, which has the added benefit of an internal heater for *very* hot water. One is a twin-tub (with American-style agitator), which I have and works great, though you have to transfer the load between wash-tub and spin-tub twice (once after wash cycle, once after rinse cycle). The one I think you'd probably want is the conventional compact top loader, which also has a regular American-style agitator. The latter two machines can be bought online for about $250 - $300. Load size is about 5 lbs., which is typical of compact machines (though the Whirlpool and Kenmore might hold more than this).
Re. "pulsators" (the spinning disc type): Some people hate 'em but some love 'em. Hoover made a line of compact washers in the 50s through the 70s that were quite popular (and are presently regarded as classics), with an impeller that was mounted in the side of the washtub. These typically did a wash cycle in *four minutes.* One version was a twin-tub, one had a wringer, and one was a stand-alone that had a matching spin-dryer. There are presently a number of Asian machines and a few European ones available in the US with pulsators (aside from Haier); some of these are apparently quite good, others not so good. Someone should do a comprehensive review of all of 'em under standard test conditions.
The key to not getting tangle troubles with a pulsator-type machine, is to load less than rated capacity, and wash mixed loads (i.e. different sized items in each load). Not more than two pairs of blue jeans at once in a compact machine, and try to avoid loads that consist only of items with long bits like pant legs and shirt sleeves.
Also: when you leave either house to go to the other, unplug the washer & dryer from the wall AC power (general appliance safety), and turn off the water outlet that supplies water to the machine. The latter point is because water hoses occasionally wear and burst from the pressure, and that could cause a major flood and much damage. Leave the lid or door of the washer open overnight after doing laundry, so the interior can dry out (prevents mold from getting started); but then close it when you leave to go to the other house (prevents stray animals from climbing in there & making nests).